# 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 ProofOfWeierstrassMtest 2013-03-22 12:58:01 2013-03-22 12:58:01 CWoo (3771) CWoo (3771) 5 CWoo (3771) Proof msc 30A99 CauchySequence