# Statement(s) > [!NOTE] Statement 1 (Weierstrass M-test) > Let $(f_{k}),f_{k}:\Omega \subseteq \mathbb{R}\to \mathbb{R}$ be a [[Sequences|sequence]] in [[Real Function|real functions]]. Assuming that for every $k$ there exists $M_{k}>0$ such that $\Vert f_{k}\Vert_{\infty}\leq M_{k}$ and $\sum_{k=1}^\infty M_{k}$ is a [[Convergent Real Series|convergent real series]], then $\sum_{k=1}^\infty f_{k}$ [[Uniform Convergence of Series of Real Functions|converges uniformly]] on $\Omega$. **Remark**: can remember as $\sum_{k=1}^\infty \lVert f _{k}\rVert_{\infty} < \infty$ then $\sum_{k=1}^{\infty} f_{k}$ converges uniformly. # Proof(s) **Proof of statement 1:** STS $S_{n}:= \sum_{k=1}^n f_{k}$ is uniformly Cauchy since [[Sequence of Real Functions is Uniformly Convergent iff Uniformly Cauchy]]. Now since $\sum_{k=1}^\infty M_{k}<\infty,$ given $\varepsilon>0,$ there exists a positive integer $N$ such that for all $n> m>N,$ $\sum_{k=m+1}^n M_{k} <\varepsilon.$Applying the triangle inequality yields then that for all $x\in \Omega,$ $\left|S_n(x)-S_m(x)\right|=\left|\sum_{k=1}^n f_k(x)-\sum_{k=1}^m f_k(x)\right|=\left|\sum_{k=m+1}^n f_k(x)\right| \leqslant \sum_{k=m+1}^n\left|f_k\right| \leqslant \sum_{k=m+1}^n M_k \leqslant \varepsilon.$Therefore $S_{n}$ is uniformly Cauchy. $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography