[求助]谁能帮我翻译一下这篇文章?~
<p>文章是讲一致收敛和微分的~我英语不好看不明白~请大家帮帮忙啊~~~~</p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><font face="Times New Roman"><b><i><span lang="EN-US">Theorem 9.13. </span></i></b><i><span lang="EN-US">Assume that each term of </span></i></font><span lang="EN-US"><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shapetype id="_x0000_t75" stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" opreferrelative="t" ospt="75" coordsize="21600,21600"><stroke joinstyle="miter"></stroke><formulas><f eqn="if lineDrawn pixelLineWidth 0"></f><f eqn="sum @0 1 0"></f><f eqn="sum 0 0 @1"></f><f eqn="prod @2 1 2"></f><f eqn="prod @3 21600 pixelWidth"></f><f eqn="prod @3 21600 pixelHeight"></f><f eqn="sum @0 0 1"></f><f eqn="prod @6 1 2"></f><f eqn="prod @7 21600 pixelWidth"></f><f eqn="sum @8 21600 0"></f><f eqn="prod @7 21600 pixelHeight"></f><f eqn="sum @10 21600 0"></f></formulas><path oconnecttype="rect" gradientshapeok="t" oextrusionok="f"></path><lock aspectratio="t" vext="edit"></lock></shapetype><shape id="_x0000_i1025" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 20.25pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image001.wmz"><font face="Times New Roman"></font></imagedata></shape></span><i><font face="Times New Roman">is a real-valued function having a finite derivative at each point of an open interval</font></i><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1026" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image003.wmz"></imagedata></font></shape></span><i><font face="Times New Roman">. Assume that for at least one point</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1027" oole="" type="#_x0000_t75" style="WIDTH: 12.75pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image005.wmz"></imagedata></font></shape></span><font face="Times New Roman">in</font></i><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1028" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image007.wmz"></imagedata></font></shape></span><i><font face="Times New Roman">the sequence </font></i><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1029" oole="" type="#_x0000_t75" style="WIDTH: 42.75pt; HEIGHT: 20.25pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image008.wmz"><font face="Times New Roman"></font></imagedata></shape></span><i><font face="Times New Roman">converges. Assume further that there exists a function g such that</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1030" oole="" type="#_x0000_t75" style="WIDTH: 39pt; HEIGHT: 18.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image010.wmz"></imagedata></font></shape></span><font face="Times New Roman">uniformly on</font></i><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1031" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image012.wmz"></imagedata></font></shape></span><i><font face="Times New Roman">. Then:<p></p></font></i></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><i><span lang="EN-US"><p><font face="Times New Roman">
</font></p></span></i></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">a) <i>There exists a function f such that</i></font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1096" oole="" type="#_x0000_t75" style="WIDTH: 39.75pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image013.wmz"></imagedata></font></shape></span><i><font face="Times New Roman">uniformly on</font></i><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1032" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image015.wmz"></imagedata></font></shape></span><font face="Times New Roman">.</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><i><span lang="EN-US"><font face="Times New Roman">b) For each x in</font></span></i><span lang="EN-US"><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1033" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image016.wmz"></imagedata></font></shape></span><i><font face="Times New Roman">the derivative</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1034" oole="" type="#_x0000_t75" style="WIDTH: 29.25pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image017.wmz"></imagedata></font></shape></span><font face="Times New Roman">exists and equal</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1035" oole="" type="#_x0000_t75" style="WIDTH: 26.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image019.wmz"></imagedata></font></shape></span><font face="Times New Roman">.<p></p></font></i></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><i><span lang="EN-US"><font face="Times New Roman">Proof. </font></span></i><span lang="EN-US"><font face="Times New Roman">Assume that</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1036" oole="" type="#_x0000_t75" style="WIDTH: 45pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image021.wmz"></imagedata></font></shape></span><font face="Times New Roman">and define a new sequence</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1037" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image023.wmz"></imagedata></font></shape></span><font face="Times New Roman">as follows:</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">
</font></p></span></p><p class="MsoNormal" align="right" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: right;"><span lang="EN-US"><span style="POSITION: relative; TOP: 24pt; mso-text-raise: -24.0pt;"><shape id="_x0000_i1038" oole="" type="#_x0000_t75" style="WIDTH: 159pt; HEIGHT: 54pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image025.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;">
</span>(8)</font></span></p><p class="MsoNormal" align="right" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: right;"><span lang="EN-US"><p><font face="Times New Roman">
</font></p></span></p><p class="MsoNormal" align="left" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left;"><span lang="EN-US"><font face="Times New Roman">The sequence</font><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1039" oole="" type="#_x0000_t75" style="WIDTH: 24.75pt; HEIGHT: 20.25pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image027.wmz"></imagedata></font></shape></span><font face="Times New Roman">so formed depends on the choice of <i>c</i>. Convergence of </font><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1040" oole="" type="#_x0000_t75" style="WIDTH: 39pt; HEIGHT: 20.25pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image029.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">follows from the hypothesis, since</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1041" oole="" type="#_x0000_t75" style="WIDTH: 66.75pt; HEIGHT: 18.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image031.wmz"></imagedata></font></shape></span><font face="Times New Roman">. We will prove next that</font><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1042" oole="" type="#_x0000_t75" style="WIDTH: 24.75pt; HEIGHT: 20.25pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image033.wmz"></imagedata></font></shape></span><font face="Times New Roman">converges uniformly on</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1043" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image034.wmz"></imagedata></font></shape></span><font face="Times New Roman">. If </font><span style="POSITION: relative; TOP: 3pt; mso-text-raise: -3.0pt;"><shape id="_x0000_i1044" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 12pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image035.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">, we have</font></span></p><p class="MsoNormal" align="left" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left;"><span lang="EN-US"><p><font face="Times New Roman">
</font></p></span></p><p class="MsoNormal" align="right" style="MARGIN: 0cm 0cm 0pt; WORD-BREAK: break-all; TEXT-ALIGN: right;"><span lang="EN-US"><span style="POSITION: relative; TOP: 12pt; mso-text-raise: -12.0pt;"><shape id="_x0000_i1045" oole="" type="#_x0000_t75" style="WIDTH: 134.25pt; HEIGHT: 30.75pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image037.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">,<span style="mso-spacerun: yes;">
</span>(9)</font></span></p><p class="MsoNormal" align="left" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left;"><span lang="EN-US"><p><font face="Times New Roman">
</font></p></span></p><p class="MsoNormal" align="left" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left;"><span lang="EN-US"><font face="Times New Roman">where</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1046" oole="" type="#_x0000_t75" style="WIDTH: 99.75pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image039.wmz"></imagedata></font></shape></span><font face="Times New Roman">. Now</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1047" oole="" type="#_x0000_t75" style="WIDTH: 27pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image041.wmz"></imagedata></font></shape></span><font face="Times New Roman">exists for each <i>x</i> in</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1048" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image043.wmz"></imagedata></font></shape></span><font face="Times New Roman">and has the value</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1049" oole="" type="#_x0000_t75" style="WIDTH: 66.75pt; HEIGHT: 18.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image044.wmz"></imagedata></font></shape></span><font face="Times New Roman">. Applying the Mean-Value Theorem in (9), we get</font></span></p><p class="MsoNormal" align="right" style="MARGIN: 0cm 0cm 0pt; WORD-BREAK: break-all; TEXT-ALIGN: right;"><span lang="EN-US"><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1050" oole="" type="#_x0000_t75" style="WIDTH: 150.75pt; HEIGHT: 18.75pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image046.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">,<span style="mso-spacerun: yes;">
</span><span style="mso-spacerun: yes;">
</span>(10)</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">where</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1051" oole="" type="#_x0000_t75" style="WIDTH: 12pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image048.wmz"></imagedata></font></shape></span><font face="Times New Roman">lies between <i>x</i> and <i>c.</i> Since</font><span style="POSITION: relative; TOP: 8pt; mso-text-raise: -8.0pt;"><shape id="_x0000_i1052" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 21.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image050.wmz"></imagedata></font></shape></span><font face="Times New Roman">converges uniformly on</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1053" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image052.wmz"></imagedata></font></shape></span><font face="Times New Roman">(by hypothesis), we can use (10), together with the Cauthy condition, to deduce that</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1054" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image053.wmz"></imagedata></font></shape></span><font face="Times New Roman">converges uniformly on</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1055" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image054.wmz"></imagedata></font></shape></span><font face="Times New Roman">.</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 21pt; mso-char-indent-count: 2.0;"><span lang="EN-US"><font face="Times New Roman">Now we can show that</font><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1056" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 20.25pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image055.wmz"></imagedata></font></shape></span><font face="Times New Roman">converges uniformly on</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1057" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image056.wmz"></imagedata></font></shape></span><font face="Times New Roman">. Let us form the particular sequence</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1058" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image053.wmz"></imagedata></font></shape></span><font face="Times New Roman">corresponding to the special point</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1059" oole="" type="#_x0000_t75" style="WIDTH: 30.75pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image057.wmz"></imagedata></font></shape></span><font face="Times New Roman">for which</font><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1060" oole="" type="#_x0000_t75" style="WIDTH: 42.75pt; HEIGHT: 20.25pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image059.wmz"></imagedata></font></shape></span><font face="Times New Roman">is assumed to converge. Form (8) we can write</font></span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 21pt; TEXT-ALIGN: center; mso-char-indent-count: 2.0;"><span lang="EN-US"><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1061" oole="" type="#_x0000_t75" style="WIDTH: 147pt; HEIGHT: 18pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image060.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">an equation which holds for every <i>x </i>in</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1062" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image062.wmz"></imagedata></font></shape></span><font face="Times New Roman">. Hence we have</font></span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US"><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1063" oole="" type="#_x0000_t75" style="WIDTH: 231pt; HEIGHT: 18pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image063.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">This equation, with the help of the Cauthy condition, establishes the uniform convergence of </font><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1064" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 20.25pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image065.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">on</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1065" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image066.wmz"></imagedata></font></shape></span><font face="Times New Roman">. This proves (a).</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">To prove (b), return to the sequence</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1066" oole="" type="#_x0000_t75" style="WIDTH: 24pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image053.wmz"></imagedata></font></shape></span><font face="Times New Roman">defined by (8) for an <i>arbitrary</i> point <i>c</i> in</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1067" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image067.wmz"></imagedata></font></shape></span><font face="Times New Roman">and let</font><span style="POSITION: relative; TOP: 10pt; mso-text-raise: -10.0pt;"><shape id="_x0000_i1068" oole="" type="#_x0000_t75" style="WIDTH: 84pt; HEIGHT: 21.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image068.wmz"></imagedata></font></shape></span><font face="Times New Roman">. The hypothesis that</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1069" oole="" type="#_x0000_t75" style="WIDTH: 14.25pt; HEIGHT: 18.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image070.wmz"></imagedata></font></shape></span><font face="Times New Roman">exists means that </font><span style="POSITION: relative; TOP: 10pt; mso-text-raise: -10.0pt;"><shape id="_x0000_i1070" oole="" type="#_x0000_t75" style="WIDTH: 86.25pt; HEIGHT: 21.75pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image072.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">. In other words, each</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1071" oole="" type="#_x0000_t75" style="WIDTH: 15pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image074.wmz"></imagedata></font></shape></span><font face="Times New Roman">is continuous at <i>c</i>. Since</font><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1072" oole="" type="#_x0000_t75" style="WIDTH: 41.25pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image076.wmz"></imagedata></font></shape></span><font face="Times New Roman">uniformly on</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1073" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image078.wmz"></imagedata></font></shape></span><font face="Times New Roman">, the limit function <i>G </i>is also continuous at <i>c</i>. This means that</font></span></p><p class="MsoNormal" align="right" style="MARGIN: 0cm 0cm 0pt; WORD-BREAK: break-all; TEXT-ALIGN: right;"><span lang="EN-US"><span style="POSITION: relative; TOP: 10pt; mso-text-raise: -10.0pt;"><shape id="_x0000_i1074" oole="" type="#_x0000_t75" style="WIDTH: 81.75pt; HEIGHT: 21.75pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image079.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;">
</span>(11)</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">the existence of the limit being part of the conclusion. But for</font><span style="POSITION: relative; TOP: 3pt; mso-text-raise: -3.0pt;"><shape id="_x0000_i1075" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 12pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image081.wmz"></imagedata></font></shape></span><font face="Times New Roman">, we have</font></span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US"><span style="POSITION: relative; TOP: 12pt; mso-text-raise: -12.0pt;"><shape id="_x0000_i1076" oole="" type="#_x0000_t75" style="WIDTH: 249pt; HEIGHT: 30.75pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image082.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">Hence, (11) states that the derivative</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1077" oole="" type="#_x0000_t75" style="WIDTH: 29.25pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image084.wmz"></imagedata></font></shape></span><font face="Times New Roman">exists and equals</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1078" oole="" type="#_x0000_t75" style="WIDTH: 26.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image086.wmz"></imagedata></font></shape></span><font face="Times New Roman">. But</font></span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US"><span style="POSITION: relative; TOP: 10pt; mso-text-raise: -10.0pt;"><shape id="_x0000_i1079" oole="" type="#_x0000_t75" style="WIDTH: 173.25pt; HEIGHT: 23.25pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image088.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US" style="mso-bidi-font-weight: bold;"><font face="Times New Roman">hence</font></span><span lang="EN-US"><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1080" oole="" type="#_x0000_t75" style="WIDTH: 62.25pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image090.wmz"></imagedata></font></shape></span><font face="Times New Roman">. Since <i style="mso-bidi-font-style: normal;">c </i>is an arbitrary point of</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1081" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image092.wmz"></imagedata></font></shape></span><font face="Times New Roman">, this proves (b).</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 21pt;"><span lang="EN-US"><font face="Times New Roman">When we reformulate Theorem 9.13 in terms of series, we obtain</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">
</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><font face="Times New Roman"><b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><span lang="EN-US">Theorem 9.14. </span></i></b><i style="mso-bidi-font-style: normal;"><span lang="EN-US">Assume that each</span></i></font><span lang="EN-US"><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1082" oole="" type="#_x0000_t75" style="WIDTH: 14.25pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image093.wmz"></imagedata></font></shape></span><i style="mso-bidi-font-style: normal;"><font face="Times New Roman">is a real-valued function defined on</font></i><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1083" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image095.wmz"></imagedata></font></shape></span><i style="mso-bidi-font-style: normal;"><font face="Times New Roman">such that the derivative</font></i><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1084" oole="" type="#_x0000_t75" style="WIDTH: 29.25pt; HEIGHT: 18.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image096.wmz"></imagedata></font></shape></span><i style="mso-bidi-font-style: normal;"><font face="Times New Roman">exists for each x in</font></i><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1085" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image098.wmz"></imagedata></font></shape></span><i style="mso-bidi-font-style: normal;"><font face="Times New Roman">. Assume that, for at least one point</font></i><span style="POSITION: relative; TOP: 6pt; mso-text-raise: -6.0pt;"><shape id="_x0000_i1086" oole="" type="#_x0000_t75" style="WIDTH: 12.75pt; HEIGHT: 18pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image099.wmz"></imagedata></font></shape></span><font face="Times New Roman">in</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1087" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image100.wmz"></imagedata></font></shape></span><i style="mso-bidi-font-style: normal;"><font face="Times New Roman">, the series</font></i><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1088" oole="" type="#_x0000_t75" style="WIDTH: 48pt; HEIGHT: 20.25pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image101.wmz"></imagedata></font></shape></span><i style="mso-bidi-font-style: normal;"><font face="Times New Roman">converges. Assume further that there exists a function g such that </font></i><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1089" oole="" type="#_x0000_t75" style="WIDTH: 78pt; HEIGHT: 20.25pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image103.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">(<i style="mso-bidi-font-style: normal;">uniformly on</i></font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1090" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image105.wmz"></imagedata></font></shape></span><font face="Times New Roman">). <i style="mso-bidi-font-style: normal;">Then:<p></p></i></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 22.5pt; TEXT-INDENT: -18pt; mso-list: l0 level1 lfo1;"><font face="Times New Roman"><i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">a)<span style="FONT: 7pt "Times New Roman";">
</span></span></span></i><i style="mso-bidi-font-style: normal;"><span lang="EN-US">There exists a function f such that</span></i></font><span lang="EN-US"><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1091" oole="" type="#_x0000_t75" style="WIDTH: 78.75pt; HEIGHT: 20.25pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image106.wmz"></imagedata></font></shape></span><font face="Times New Roman">(uniformly on</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1092" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image108.wmz"></imagedata></font></shape></span><font face="Times New Roman">).<i style="mso-bidi-font-style: normal;"><p></p></i></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 22.5pt; TEXT-INDENT: -18pt; mso-list: l0 level1 lfo1;"><font face="Times New Roman"><i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">b)<span style="FONT: 7pt "Times New Roman";">
</span></span></span></i><i style="mso-bidi-font-style: normal;"><span lang="EN-US">If </span></i></font><span lang="EN-US"><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1093" oole="" type="#_x0000_t75" style="WIDTH: 45.75pt; HEIGHT: 15.75pt;"><imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image109.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman">, the derivative</font><span style="POSITION: relative; TOP: 5pt; mso-text-raise: -5.0pt;"><shape id="_x0000_i1094" oole="" type="#_x0000_t75" style="WIDTH: 30pt; HEIGHT: 15.75pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image111.wmz"></imagedata></font></shape></span><font face="Times New Roman">exists and equals</font><span style="POSITION: relative; TOP: 7pt; mso-text-raise: -7.0pt;"><shape id="_x0000_i1095" oole="" type="#_x0000_t75" style="WIDTH: 44.25pt; HEIGHT: 20.25pt;"><font face="Times New Roman">
<imagedata otitle="" src="file:///C:\DOCUME~1\cps\LOCALS~1\Temp\msohtmlclip1\01\clip_image113.wmz"></imagedata></font></shape></span><font face="Times New Roman">.<i style="mso-bidi-font-style: normal;"><p></p></i></font></span></p><p></p> 我怎么只看到脚本文件呢？ “<font new="" roman="" face="Times new="><b><i><span lang="EN-US">Theorem 9.13. </span></i></b><i><span lang="EN-US">Assume that each term of </span></i></font><span lang="EN-US"><span style=""><shapetype id="_x0000_t75" stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" opreferrelative="t" ospt="75" coordsize="21600,21600"><stroke joinstyle="miter"></stroke><formulas><f eqn="if linedrawn=" linedrawn="" pixellinewidth="pixelLineWidth" 0=""></f><f eqn="sum @0 1=" 1="" 0=""></f><f eqn="sum 0=" 0="" @1=""></f><f eqn="prod @2 1=" 1="" 2=""></f><f eqn="prod @3 21600=" 21600="" pixelwidth=""></f><f eqn="prod @3 21600=" 21600="" pixelheight=""></f><f eqn="sum @0 0=" 0="" 1=""></f><f eqn="prod @6 1=" 1="" 2=""></f><f eqn="prod @7 21600=" 21600="" pixelwidth=""></f><f eqn="sum @8 21600=" 21600="" 0=""></f><f eqn="prod @7 21600=" 21600="" pixelheight=""></f><f eqn="sum @10 21600=" 21600="" 0=""></f></formulas><path oc="oc" gradientshapeok="t" oextrusi="oextrusi"></path><lock aspectratio="t" vext="edit"></lock></shapetype><shape id="_x0000_i1025" oole="" type="#_x0000_t75" style="width: 24pt; height: 20.25pt;"><imagedata otitle="" src="file:///C:%5CDOCUME%7E1%5Ccps%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_image001.wmz"></imagedata></shape></span><i><font new="" roman="" face="Times new=">is a。。。。。” 这里面有没有漏了字？<br/></font></i></span>
[此贴子已经被作者于2008-7-28 14:24:38编辑过] 看不清楚。。。。。
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