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 $p\le q$,then, for every $x\in X$, we have

$|s_q(x)-s_p(x)|$$=$$\left|{\displaystyle \sum _{m=p+1}^q}{f}_m(x)\right|$

$\le$${\displaystyle \sum _{m=p+1}^q}|f_m(x)|$

$\le$${\displaystyle \sum _{m=p+1}^q}{M}_m$

But since ${\sum }_{n=1}^{\mathrm{\infty }}{M}_n$ converges, for any $ϵ>0$ we canfind an $N\in \mathbb{N}$ such that, for any $p,q>N$ and $x\in X$, we have. Hence the sequence$s_n$ converges uniformly to ${\sum }_{n=1}^{\mathrm{\infty }}{f}_n$.