# Math Help - Prove this series of functions convergent uniformly

1. ## Prove this series of functions convergent uniformly

$(u_{n}(x) : n=0,1,...)$ be a sequence of real-valued functions on subset E of R.

suppose for all $x\in E, |u_{n}(x)|\leq M_{n}$

where $\sum_{n=0}^{\infty}{M_{n}}$ converges.

prove that: $\sum_{0}^{\infty}{u_{n}(x)}$ converges uniformly on E.

2. Originally Posted by szpengchao
$(u_{n}(x) : n=0,1,...)$ be a sequence of real-valued functions on subset E of R.

suppose for all $x\in E, |u_{n}(x)|\leq M_{n}$

where $\sum_{n=0}^{\infty}{M_{n}}$ converges.

prove that: $\sum_{0}^{\infty}{u_{n}(x)}$ converges uniformly on E.
A sequence of real-valued functions on E is uniformly convergent if and only if it is uniformly Cauchy.
That is a fact used in the link Opalg gave above.