There is N so that[tex] |S_n(x) - S_m(x)| \leq \epsilon [/tex] for

[Pearce_09]

there is N so that
|S_n(x) - S_m(x)| \leq \epsilon for ever x in I if n,m \geq N

( prove by cauchy's criterion )

claim: lim S_n(x) = S(x)

|S_n(x) - S(x)| < \epsilon /2 if n\geq N
then,
|S_n(x) - S_m(x)| < |S_n(x) - S(x)| + |S(x) - S_m(x)|
< \epsilon /2 + \epsilon /2
< \epsilon

therefor the series converges pointwise to a funtion S(x)
... and I am not sure how to show that this converging uniformly on I

 

[TD]

How did you define uniform convergence?

I have that a sequence of functions s_n \left( x \right) converges uniformely to the function \ell :\left( {a,b} \right) \to \mathbb{R} if:

\forall \varepsilon &gt; 0,\exists n &gt; N \Rightarrow \forall x \in \left( {a,b} \right):\left| {s_n \left( x \right) - \ell \left( x \right)} \right| &lt; \varepsilon

If I understand, you need to show that a sequence of functions s_n \left( x \right) converges uniformely over that interval iff (I believe you only have to prove it one way, using this to prove uniform convergence) it satisfies Cauchy's criterion, being:

\forall \varepsilon &gt; 0,\exists N:m,n &gt; N \Rightarrow \forall x \in \left( {a,b} \right):\left| {s_n \left( x \right) - s_m \left( x \right)} \right| &lt; \varepsilon (*)

So assume that \left( {s_n } \right) satisfies (*). Then for all x in I, the numerical sequence \left( {s_n \left( x \right)} \right) is a Cauchy-sequence and is therefore convergent, call the limit l(x). So we already have pointwise convergence to l(x), the only thing you need to show now is that the convergence is uniform. Now use the assumption of Cauchy's criterion.