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-1 241 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle26" -1 242 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle12" -1 243 "Times" 1 14 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 203 "" 0 "" {TEXT 239 0 "" }{TEXT 239 1 "\n" } {TEXT 239 8 " " }{TEXT 239 0 "" }{TEXT 239 1 "\n" }{TEXT 239 21 "Pointwise Convergence" }{TEXT 239 0 "" }{TEXT 239 1 "\n" }{TEXT 239 8 " " }{TEXT 239 0 "" }{TEXT 239 1 "\n" }{TEXT 239 22 "Poin twise Convergence." }{TEXT 239 0 "" }}{PARA 238 "" 0 "" {TEXT 244 13 " A sequence \{ " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\" ), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-m i(\"n\", italic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"), Typesetting:-mo(\"⁡\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\")), Typesetting:- mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,mathvaria ntGQ'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$6-Q0& ApplyFunction;F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stret chyGFM/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'ls paceGQ&0.0emF'/%'rspaceGFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F7F:FIF+" } {TEXT 244 45 "\} of functions from a set S to R is said to " }{TEXT 215 18 "converge pointwise" }{TEXT 244 52 " to a function f if for e very x in S we have \{" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesett ing:-mi(\"\"), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"f \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Ty pesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), su bscriptshift = \"0\"), Typesetting:-mo(\"⁡\", mathvarian t = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\" ), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", itali c = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\")), \+ Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_ syslibGF'6%-I#miGF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"fF'/%'italicGQ%trueF '/%,mathvariantGQ'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F' -I#moGF$6-Q0⁡F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separat orGFM/%)stretchyGFM/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'a ccentGFM/%'lspaceGQ&0.0emF'/%'rspaceGFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF 'F7F:FIF+" }{TEXT 244 54 "\} converges to f(x) as n tends to infini ty. " }{TEXT 244 0 "" }{TEXT 244 1 "\n" }}{PARA 238 "" 0 "" {TEXT 218 19 "Bounded Convergence" }{TEXT 244 0 "" }{TEXT 244 1 "\n" } {TEXT 244 8 " " }{TEXT 244 0 "" }{TEXT 244 1 "\n" }{TEXT 244 12 "A sequence \{" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"f\", ital ic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesettin g:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), subscripts hift = \"0\"), Typesetting:-mo(\"⁡\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"tr ue\", mathvariant = \"italic\")), mathvariant = \"normal\")), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,math variantGQ'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$ 6-Q0⁡F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%) stretchyGFM/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM /%'lspaceGQ&0.0emF'/%'rspaceGFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F7F:FIF +" }{TEXT 244 49 "\} of functions from a set S to R is said to " }{TEXT 215 18 "converge boundedly" }{TEXT 244 22 " to a function f if \{" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesett ing:-mrow(Typesetting:-msub(Typesetting:-mi(\"f\", italic = \"true\", \+ mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"n\", it alic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"), \+ Typesetting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced (Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvaria nt = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")); " "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q! F'-F#6%-I%msubGF$6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,mathvariantGQ'itali cF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$6-Q0&ApplyFunct ion;F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*s ymmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceGQ&0.0 emF'/%'rspaceGFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F7F:FIF+" }{TEXT 244 82 "\} converges pointwise to f and it is possible to find a number K such that |" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"f\", ital ic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesettin g:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), subscripts hift = \"0\"), Typesetting:-mo(\"⁡\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"tr ue\", mathvariant = \"italic\")), mathvariant = \"normal\")), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,math variantGQ'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$ 6-Q0⁡F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%) stretchyGFM/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM /%'lspaceGQ&0.0emF'/%'rspaceGFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F7F:FIF +" }{TEXT 244 45 "| <= K for every x in S and every n. " } {TEXT 244 0 "" }{TEXT 244 1 "\n" }}{PARA 238 "" 0 "" {TEXT 218 19 "Uni form Convergence" }{TEXT 244 0 "" }{TEXT 244 1 "\n" }{TEXT 244 8 " \+ " }{TEXT 244 0 "" }{TEXT 244 1 "\n" }{TEXT 244 12 "A sequence \{" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- mrow(Typesetting:-msub(Typesetting:-mi(\"f\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic \+ = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"), Types etting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Type setting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \+ \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F# 6%-I%msubGF$6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F #6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$6-Q0⁡F '/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*symmet ricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceGQ&0.0emF'/ %'rspaceGFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F7F:FIF+" }{TEXT 244 48 "\} of functions from a set S to R is said to " }{TEXT 215 18 "conver ge uniformly" }{TEXT 244 28 " to a function f if sup |" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Types etting:-msub(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \"true \", mathvariant = \"italic\")), subscriptshift = \"0\"), Typesetting:- mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:- mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic \")), mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6# /I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-I%msu bGF$6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-F,6% Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$6-Q0⁡F'/F;Q'no rmalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*symmetricGFM/% (largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceGQ&0.0emF'/%'rspace GFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F7F:FIF+" }{TEXT 244 3 " - " }{TEXT 230 4 "f(x)" }{TEXT 244 83 " | tends to 0 as n tends to infinity. \+ This is equivalent to say that for every " }}{PARA 238 "" 0 "" {TEXT 244 114 "epsilon >0, there exists a natural number N such that for all x in S, and for all n is larger than or equal to N, " }{TEXT 244 1 "| " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -mrow(Typesetting:-msub(Typesetting:-mi(\"f\", italic = \"true\", math variant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"), Type setting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Type setting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \+ \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F# 6%-I%msubGF$6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F #6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$6-Q0⁡F '/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*symmet ricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceGQ&0.0emF'/ %'rspaceGFfn-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F7F:FIF+" }{TEXT 244 3 " - " }{TEXT 230 4 "f(x)" }{TEXT 244 12 " |< epsilon." }{TEXT 244 7 " \+ " }{TEXT 244 0 "" }{TEXT 244 1 "\n" }}{PARA 238 "" 0 "" {TEXT 215 10 "Exercise 1" }{TEXT 244 103 ": Demonstrate that the sequence of fun ctions f(x,n)=(n*x)/(1+(n^2)*x^2) is pointwise convergent to 0." } {TEXT 244 0 "" }}{PARA 213 "> " 0 "" {MPLTEXT 1 216 37 "f:=proc(x,n) ( n*x)/(1+(n^2)*x^2) end;" }{MPLTEXT 1 216 0 "" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6$I\"xG6\"I\"nG6\"6\"6\"6\"*(I\"nG6\"\"\"\"I \"xG6\"\"\"\",&\"\"\"\"\"\"*&)I\"nG6\"\"\"#\"\"\")I\"xG6\"\"\"#\"\"\" \"\"\"!\"\"6\"6\"6\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 216 7 "f(x,n);" }{MPLTEXT 1 216 0 "" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#*(I\"nG6\"\"\"\"I\"xG6\"\"\"\",&\"\"\"\"\"\"*&)I\"nG6\" \"\"#\"\"\")I\"xG6\"\"\"#\"\"\"\"\"\"!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 216 70 "plot(\{f(x,10^5),f(x,10^6),f(x, 10^7)\},x=-10^(-5)..10^(-5),thickness=2);" }{MPLTEXT 1 216 0 "" }} {PARA 234 "" 1 "" {TEXT 245 0 "" }{GLPLOT2D 400 400 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G%+HORIZONTALG-%*LINESTYLEG6#\"\"!-%*AXESSTYLEG6#%'NORMALG-%*GRIDSTYLE G6#%,RECTANGULARG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" " Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12 " "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "C urve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" }}{TEXT 209 0 "" }}}{EXCHG {PARA 222 "" 0 "" {TEXT 242 10 "Exercise 2 " }{TEXT 247 59 ": Demonstrate by animation that the sequence of funct ions " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Types etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi( \"f\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"⁡\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mro w(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", s eparator = \"true\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\")), T ypesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typese tting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\"), Typeset ting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", ital ic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&Invisible Times;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-mo(\"&Exponential E;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.1111111em\"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0; \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-msup(Typese tting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\"), Typeset ting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\"), \+ Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x \", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\" \")), superscriptshift = \"0\")), Typesetting:-mi(\"\")), Typesetting: -mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%- I#miGF$6#Q!F'-F#6'F+-F#6%-F,6%Q\"fF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'-I#moGF$6-Q0⁡F'/F:Q'normalF'/%&fenceGQ&falseF'/ %*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimi tsGFD/%'accentGFD/%'lspaceGQ&0.0emF'/%'rspaceGFS-I(mfencedGF$6$-F#6%-F 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