I am asked to prove the following:
Let $\sum_{n=1}^{\infty} f_n$ be a series of continuous functions in an set $K \subset \mathbb{R}^p$. If there exists constants $a_n > 0,\forall n \in \mathbb{N}$ such that $\sum_{n=1}^{\infty} a_n$ converges, and $|f_n(x)|<a_n$ for $\forall n \in \mathbb{N}$ and $\forall x \in K$, then $\sum_{n=1}^{\infty} f_n$ is converges uniformly in $K$.
My approach:
If $|f_n(x)|<a_n$ and because $\sum_{n=1}^{\infty} a_n$ converges, then $\sum_{n=1}^{\infty} |f_n|$ converges as well. This makes $\sum_{n=1}^{\infty} f_n$ absolutely convergent. But how can I show that this convergence is uniform and not pointwise?
You have established $\sum_{n \geqslant 1} f_n$ converges; call its limit $ F(x) $. Then for $ x \in K $, use the convergence of $\sum_{n \geqslant 1} a_n $ to choose $N$ and calculate a bound for $$\left| F(x) - \sum_{n = 1}^{N} f_n(x) \right|. $$ You should find the same bound works for all $x$.