This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2003.2.18) 20 OCT 2003 11:12 **top_7 (top_7.tex LaTeX2e <2001/06/01> Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. (/usr/local/share/texmf/tex/latex/base/article.cls Document Class: article 2001/04/21 v1.4e Standard LaTeX document class (/usr/local/share/texmf/tex/latex/base/size10.clo File: size10.clo 2001/04/21 v1.4e Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) (/usr/local/share/texmf/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. (/usr/local/share/texmf/tex/latex/amsmath/amstext.sty Package: amstext 2000/06/29 v2.01 (/usr/local/share/texmf/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 \@emptytoks=\toks14 \ex@=\dimen103 )) (/usr/local/share/texmf/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d \pmbraise@=\dimen104 ) (/usr/local/share/texmf/tex/latex/amsmath/amsopn.sty Package: amsopn 1999/12/14 v2.01 operator names ) \inf@bad=\count87 LaTeX Info: Redefining \frac on input line 211. \uproot@=\count88 \leftroot@=\count89 LaTeX Info: Redefining \overline on input line 307. \classnum@=\count90 \DOTSCASE@=\count91 LaTeX Info: Redefining \ldots on input line 379. LaTeX Info: Redefining \dots on input line 382. LaTeX Info: Redefining \cdots on input line 467. \Mathstrutbox@=\box26 \strutbox@=\box27 \big@size=\dimen105 LaTeX Font Info: Redeclaring font encoding OML on input line 567. LaTeX Font Info: Redeclaring font encoding OMS on input line 568. \macc@depth=\count92 \c@MaxMatrixCols=\count93 \dotsspace@=\muskip10 \c@parentequation=\count94 \dspbrk@lvl=\count95 \tag@help=\toks15 \row@=\count96 \column@=\count97 \maxfields@=\count98 \andhelp@=\toks16 \eqnshift@=\dimen106 \alignsep@=\dimen107 \tagshift@=\dimen108 \tagwidth@=\dimen109 \totwidth@=\dimen110 \lineht@=\dimen111 \@envbody=\toks17 \multlinegap=\skip44 \multlinetaggap=\skip45 \mathdisplay@stack=\toks18 LaTeX Info: Redefining \[ on input line 2666. LaTeX Info: Redefining \] on input line 2667. ) (/usr/local/share/texmf/tex/latex/amsfonts/amssymb.sty Package: amssymb 2002/01/22 v2.2d (/usr/local/share/texmf/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2001/10/25 v2.2f \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 132. )) (/usr/local/share/texmf/tex/latex/jknappen/mathrsfs.sty Package: mathrsfs 1996/01/01 Math RSFS package v1.0 (jk) \symrsfs=\mathgroup6 ) No file top_7.aux. \openout1 = `top_7.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 12. LaTeX Font Info: ... okay on input line 12. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 12. LaTeX Font Info: ... okay on input line 12. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 12. LaTeX Font Info: ... okay on input line 12. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 12. 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(/usr/local/share/texmf/tex/latex/jknappen/ursfs.fd File: ursfs.fd 1998/03/24 rsfs font definition file (jk) ) [1 ] [2] Overfull \hbox (3.7387pt too wide) in paragraph at lines 93--96 \OT1/cmr/m/n/10 Since $\OML/cmm/m/it/10 d[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a[ ]; x[]\OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 < ^^N[]$ \OT1/cmr/m/n/10 as was shown, we have from above that $\OML/cmm/m/it/10 d[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; g\OT 1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )) \OML/cmm/m/it/10 < [] [3] Overfull \hbox (32.72758pt too wide) in paragraph at lines 104--106 \OT1/cmr/m/n/10 Now, since $\OML/cmm/m/it/10 d[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/ 10 a[]; b[]\OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 < ^^N[]$ \OT1/cmr/m/n/10 as was s hown, we have from above that $\OML/cmm/m/it/10 d[]\OT1/cmr/m/n/10 (\OML/cmm/m/ it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b[]\OT1/cmr/m/n/10 )) \OML/cmm/m/it/10 < [] Overfull \hbox (0.5314pt too wide) in paragraph at lines 128--136 \OT1/cmr/m/n/10 Choose a point $\OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 2 \OML/cm m/m/it/10 X$ \OT1/cmr/m/n/10 and de-fine $\OML/cmm/m/it/10 x[] \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )\OML/ cmm/m/it/10 :$ \OT1/cmr/m/n/10 For $\OML/cmm/m/it/10 n > m$ \OT1/cmr/m/n/10 we have $\OML/cmm/m/it/10 d\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]; x[]\OT1/cmr/m/n/ 10 ) = [] [4] [5] [6] Overfull \hbox (1.2242pt too wide) in paragraph at lines 217--219 \OT1/cmr/m/n/10 Then for $\OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ^^T \OML/cmm/m/it /10 N$ \OT1/cmr/m/n/10 there is a neigh-bor-hood $\OML/cmm/m/it/10 U[]$ \OT1/cm r/m/n/10 of $\OML/cmm/m/it/10 x[]$ \OT1/cmr/m/n/10 such that $\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 U[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^Z \OML/cmm/m/it/10 B\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 x[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; ^^O\OT1/cmr/m/n/10 )\OML/cmm/m/i t/10 ;$ [] Overfull \hbox (23.73918pt too wide) in paragraph at lines 222--224 \OT1/cmr/m/n/10 Since $\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 U\ OT1/cmr/m/n/10 )$ is bounded by $\OML/cmm/m/it/10 M[]$ \OT1/cmr/m/n/10 this is a con-tra-dic-tion, so we must have $\OML/cmm/m/it/10 d\OT1/cmr/m/n/10 (\OML/cm m/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 )) \OML/cmm/m/it/10 < [] [7] (top_7.aux) ) Here is how much of TeX's memory you used: 1128 strings out of 20880 11948 string characters out of 196184 63158 words of memory out of 350001 4101 multiletter control sequences out of 10000+15000 11281 words of font info for 47 fonts, out of 400000 for 1000 14 hyphenation exceptions out of 10000 27i,8n,24p,759b,248s stack positions out of 3000i,100n,1500p,50000b,4000s Output written on top_7.dvi (7 pages, 35428 bytes).