How does one prove $\left\lVert\sum\limits_{n=1}^{\infty}f_n\right\rVert\le \sum\limits_{n=1}^{\infty}\lVert f_n\rVert$? Can the triangle inequality for norms works for infinite some as well?
Hope that someone can give some hints to prove it or give a counterexample. Thanks!
 A: For each $k$, $$\left\lVert\sum_{n=1}^{k}f_n\right\rVert \leqslant \sum_{n=1}^{k}\lVert f_n \rVert$$
Then take $k\to\infty$.
A: The Banach function spaces may be a better choice for your question. You can see here for the definition of a Banach function space or this book Function Spaces, Volume 1 (chapter 6).
Answer to your question:
The first thing is to introduce the definition of the Riesz-Fischer property. 

We say that  a normed linear space $(X,\left\|\cdot\right\|)$ has the
  Riesz-Fischer property if for each sequence $\{f_n\}^{\infty}_{n=1}$
  in $X$ such that  $$ \sum^{\infty}_{n=1}\left\|f_n\right\|<\infty, $$
  there exists an element $f\in X$ such that $\sum^{\infty}_{n=1}f_n=f$
  in $X$, that is,  $$ \lim_{n\rightarrow
\infty}\left\|\sum^{n}_{k=1}f_k-f\right\|=0. $$

Since every Banach function space $X$ has the Riesz-Fischer property, we have 
$$
\left\|f\right\|-\sum^n_{k=1}\left\|f_k\right\| \leq \left \|f\right \|-\left \|\sum^{n}_{k=1}f_k \right \| \leq \left \| f - \sum^{n}_{k=1}f_{k} \right\|,
$$
where $\sum^{\infty}_{n=1}\left\|f_n\right\|<\infty$, $f=\sum^\infty_{n=1}f_n$, and $f\in X$.
Then let $n \rightarrow \infty$, we get this result. For a particular example, the Lebesgue space $L^{p}(\Omega)$ is a Banach function space, where $1\leq p \leq \infty$ and $\Omega\subseteq  \mathbb{R}^{N}$ is a suitable domain. As a conclusion, the inequality $\left\|f\right\|\leq\sum^\infty_{n=1}\left\|f_n\right\| $ is still valid in $L^p(\Omega)$.

Supplement:
The inequality $\left\|f\right\|\leq\sum^\infty_{n=1}\left\|f_n\right\|$ may be invalid if $\sum^{\infty}_{n=1}\left\|f_n\right\|=\infty$ or the normed linear space $(X,\left\|\cdot\right\|)$ doesn't satisfy the Riesz-Fischer property. For the first case, we may not have the conclusion that this series $\sum^{\infty}_{n=1}f_n$ is convergent in $X$. The second is obvious.
