Analogous for Weierstrass' theorem for functional series. The Weierstrass' theorem for functional series say the following: > Suppose $\\{f_n\\}$ is a sequence of functions defined on $E$, and suppose $$|f_n(x)|\leq M_n$$ for every $x\in E$ and $n=1,2,3,...$. Then $\sum f_n$ converges uniformly on $E$ if $\sum M_n$ converges. There exist an analogous theorem for sequence of function instead of series? Thanks a lot!!!

If the sequence of functions $(f_n)$ is point-wise convergent to a function $f$ on an interval $I$ and if there's a sequence $\mu_n$ convergent to zero such that

$$|f_n(x)-f(x)|\le \mu_n,\quad \forall x\in I$$ then the convergence of $(f_n)$ is uniform.