Prove this series of functions convergent uniformly

**szpengchao** · October 25th 2008, 01:18 PM

$(u_{n}(x) : n=0,1,...)$ 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.

**Opalg** · October 25th 2008, 01:22 PM

PlanetMath: proof of Weierstrass M-test

**ThePerfectHacker** · October 25th 2008, 04:11 PM

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.

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