Prove the triangle inequality for series, that is if $\sum x_n$ converges absolutely then $|\sum_{n=1}^{\infty}x_n| \ge \sum_{n=1}^{\infty}|x_n| $.
My attempy: Since $\sum x_n$ converges absolutely, it is cauchy series. Then, $\forall \varepsilon>0,$ there exists $M\in Z^+$ such that $|\sum_{j=n+1}^{k}|x_n||<\varepsilon$ for $n\ge M$ and $k>n$. Then,
$|\sum_{j=n+1}^{k}|x_n||=\sum_{j=n+1}^{k}|x_n| \ge |\sum_{j=n+1}^{k}x_n|$ by triangle inequality.
I think it is wrong since I use triangle inequality, which is required to be proven, and I also don't know how to change $\sum_{j=n+1}^{k} \rightarrow \sum_{j=1}^{\infty}$.
Could you give some help?
Thank you in advance.
Best Answer
we have
$|\sum_{j=1}^{n}x_j|\leq \sum_{j=1}^{n}|x_j|$ by the triangle inequality wich is true for any $n \in\mathbb{N}$ so by taking $ n \rightarrow +\infty $ you can deduce $|\sum_{n=1}^{\infty}x_n| \leq \sum_{n=1}^{\infty}|x_n|$