Series of Functions

**ajj86** · January 26th 2009, 05:25 PM

This is number 2 of the problem set for real analysis problems:

Prove that if
$<br /> \sum_{n=1}^{\infty}{{g_n}}<br />$
converges uniformly, then (g_n) converges uniformly to zero.

**ajj86** · January 30th 2009, 09:19 AM

Intuitively this makes sense to me. If something converges, then that means the terms eventually get small enough so that they are infinitely small. I just can't put it into formal terms through a proof. Any ideas?

**ThePerfectHacker** · January 30th 2009, 10:17 AM

Originally Posted by ajj86

This is number 2 of the problem set for real analysis problems:

Prove that if
$<br /> \sum_{n=1}^{\infty}{{g_n}}<br />$
converges uniformly, then (g_n) converges uniformly to zero.

If $\sum_{n=1}^{\infty} g_n$ converges uniformly then it is uniformly Cauchy.
Let $\epsilon > 0$ then there is $N$ so that if $n,m>N$
Then, $\left| \sum_{k=1}^n g_k(x) - \sum_{k=1}^m g_k(x) \right| < \epsilon$ for $x\in S$ .
Thus, for example if $n=m+1$ then, $|g_n(x)| < \epsilon$ .
Thus, we see that $g_n \to 0$ uniformly on $S$ .

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