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 canfind 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 $\\sum_{n=1}^{\\infty}f_{n}$ .