$\sum_{n=1}^\infty a_n$ is absolutely convergent. Let $b_{n}$ be a subsequence of $a_n$. Prove that $\sum_{n=1}^\infty b_{n}$ converges.
$\sum_{n=1}^\infty |a_n|$ convergent $\rightarrow$ $\sum_{n=1}^\infty a_n$ convergent $\rightarrow$ $\{a_n\}_{n=1}^\infty$ convergent $\rightarrow \{b_{n}\}_{n=1}^\infty$ convergent
$\rightarrow b_{n}$ cauchy sequence $\rightarrow$ $\forall \varepsilon>0, \exists M$ s.t. $|b_m-b_n|< \varepsilon$ for $m\ge n \ge M$ $\rightarrow |\sum_{i=n+1}^mb_i| < \varepsilon$
I think this step is wrong because $\{1/n\}_{n=1}^\infty$ is cauchy sequence, but $|\sum_{i=n+1}^m 1/n| \not< \varepsilon$.
How can I proceed from here?
Thank you in advance.
Hint. If $(b_{n})_n$ is a subsequence of $(a_n)_n$ then there is a strictly increasing function $f$ from $\mathbb{N}^+$ to $\mathbb{N}^+$ such that $b_n=a_{f(n)}$. Then for any positive integer $N$, $f(N)\geq N$ and $$\sum_{n=1}^N |b_{n}|=\sum_{n=1}^N |a_{f(n)}|\leq \sum_{n=1}^{f(N)} |a_{n}|\leq \sum_{n=1}^\infty |a_n|\in \mathbb{R}.$$ What may we conclude about the absolute convergence of $\sum_{n=1}^\infty b_{n}$?