Question:
Prove that if the series $\sum_{i=1}^\infty u_n(x)$ converges uniformly on an interval, then $\lim_{n \to \infty} u_n = 0$ uniformly on that interval.
What I have so far:
We are given:
$\forall \epsilon>0 \quad \forall x \in I \quad \exists N$ s.t. $\forall n \geq N \quad |(\sum_{k=1}^n u_k) - U| < \epsilon$
where $I$ is an interval and $U := \sum_{k=1}^\infty u_k$.
We also know that: $\sum_{k=1}^\infty u_k$ converges $\implies \lim_{n \to \infty} u_n = 0$
So $\forall \epsilon>0 \quad \exists N$ s.t. $\forall n \geq N \quad |u_n - 0| < \epsilon$.
Then
$|u_n| = |u_n + \sum_{k=1}^n u_k - \sum_{k=1}^n u_k|$
$= |(\sum_{k=1}^n u_k) - u_n - \sum_{k=1}^n u_k| \quad$ (factoring out $-1$)
$= |\sum_{k=1}^{n-1} u_k - \sum_{k=1}^n u_k|$
$\leq |\sum_{k=1}^{n-1} u_k| + |\sum_{k=1}^n u_k|$
$< \epsilon + \epsilon$
$= 2 \epsilon$
Is this useful in any way, or am I completely off? I know I have to prove that $u_n$'s convergence is uniform, but I'm not quite sure how.