I'm unable to find a rigorous proof for the following problem. I've been stuck and don't know how to approach this problem.

Question

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence such that $\liminf |a_n| = 0$. Prove there is a subsequence $ (a_{n_k})_{k \in \mathbb{N}} $ such that $\sum_{k = 1}^{\infty}(a_{n_k})$ converges.

Answer 1

We first set $n_0=1$ and $c_1=1$. By the property of $\liminf$, there exists $n_1>n_0=1$ such that $|a_{n_1}|<c_1=1$.

Then we set $c_2=1/4$. Again, there exists $n_2>n_1$ such that $|a_{n_2}|<c_2=1/4$.

We can continue this fashion. At the $k$-th step, we set $c_k=1/k^2$ and we can then find some $n_k>n_{k-1}$ such that $|a_{n_k}|<c_k=1/k^2$.

Note that $$\sum_{k=1}^\infty{|a_{n_k}|}<\sum_{k=1}^\infty{\frac{1}{k^2}}.$$

Since the RHS converges, the LHS converges, i.e., $\sum_{k=1}^\infty{a_{n_k}}$ converges absolutely.