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.


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