I am working on an exercise that goes like this: consider the functions $f_n:[0,1]\rightarrow\mathbb{R}$ for $n\in\mathbb{N}$ such that they are continuous and that $\sum_{n=1}^{\infty}f_n(x)$ converges uniformly on $[0,1)$. Prove that $\sum_{n=1}^{\infty}f_n(1)$ converges.
I'm not entirely sure how one could prove this. I am aware of uniform convergence for sequences of functions, but in this case I could use some help outlining the method of the proof because I am not sure how to use the uniform convergence of series of $f_n(x)$ to prove the claim. Thank you.
$\forall \varepsilon >0, \exists N>0,\forall n>N, \forall p\in \mathbb Z^{+},\forall x\in [0,1)$, $$|f_{n+1}(x)+\ldots + f_{n+p}(x)|<\varepsilon.$$ Due to the continuity of $f_{n}(x)$, we let $x$ tend towards $1$, to obtain $$|f_{n+1}(1)+\ldots + f_{n+p}(1)|\leq\varepsilon<2\varepsilon.$$ According to Cauchy criterion,$$\sum^{+\infty}_{n=1}f_{n}(1)$$converges.