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there is N so that

[tex] |S_n(x) - S_m(x)| \leq \epsilon [/tex] for ever x in I if n,m [tex]\geq[/tex] N

( prove by cauchy's criterion )

claim: [tex] lim S_n(x) = S(x) [/tex]

[tex] |S_n(x) - S(x)| < \epsilon /2[/tex] if n[tex] \geq N [/tex]

then,

[tex] |S_n(x) - S_m(x)| < |S_n(x) - S(x)| + |S(x) - S_m(x)| [/tex]

< [tex] \epsilon /2 + \epsilon /2 [/tex]

< [tex] \epsilon [/tex]

therefor the series converges pointwise to a funtion S(x)

..... and im not sure how to show that this converging uniformly on I

[tex] |S_n(x) - S_m(x)| \leq \epsilon [/tex] for ever x in I if n,m [tex]\geq[/tex] N

( prove by cauchy's criterion )

claim: [tex] lim S_n(x) = S(x) [/tex]

[tex] |S_n(x) - S(x)| < \epsilon /2[/tex] if n[tex] \geq N [/tex]

then,

[tex] |S_n(x) - S_m(x)| < |S_n(x) - S(x)| + |S(x) - S_m(x)| [/tex]

< [tex] \epsilon /2 + \epsilon /2 [/tex]

< [tex] \epsilon [/tex]

therefor the series converges pointwise to a funtion S(x)

..... and im not sure how to show that this converging uniformly on I

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