proof of Weierstrass M-test

Consider the sequence of partial sums $s_{n}=\sum_{m=1}^{n}f_{m}$ . Take any $p,q\in\mathbb{N}$ such that $p\leq q$ ,then, for every $x\in X$ , we have

$\displaystyle\|s_{q}(x)-s_{p}(x)\|$	$\displaystyle=$	$\displaystyle\left\|\sum_{m=p+1}^{q}f_{m}(x)\right\|$
	$\displaystyle\leq$	$\displaystyle\sum_{m=p+1}^{q}\|f_{m}(x)\|$
	$\displaystyle\leq$	$\displaystyle\sum_{m=p+1}^{q}M_{m}$

But since $\sum_{n=1}^{\infty}M_{n}$ converges, for any $\epsilon>0$ we can find an $N\in\mathbb{N}$ such that, for any $p,q>N$ and $x\in X$ , we have $|s_{q}(x)-s_{p}(x)|\leq\sum_{m=p+1}^{q}M_{m}<\epsilon$ . Hence the sequence $s_{n}$ converges uniformly to $\sum_{n=1}^{\infty}f_{n}$ .

Title	proof of Weierstrass M-test
Canonical name	ProofOfWeierstrassMtest
Date of creation	2013-03-22 12:58:01
Last modified on	2013-03-22 12:58:01
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Proof
Classification	msc 30A99
Related topic	CauchySequence