Understanding properties of a sequence, $(a_n)$ given $\;\liminf|a_n|=0$

Question

I am working on proving the following. Let $(a_n)$ be a sequence such that $\;\liminf |a_n|=0$. Prove there is a subsequence $(a_{n_k})$ such that $\sum_{k=1}^{\infty}a_{n_k}$ converges.

I was thinking that eventually I want to show that this series meets the Cauchy criterion where

$\forall\epsilon>0, \exists N\in\mathbb{N}$ such that $\forall t\geq r>N, |\sum_{k=1}^{t}a_{n_k}-\sum_{k=1}^{r}a_{n_k}| = |\sum_{k=r+1}^{t}a_{n_k}|<\epsilon$

First I need figure out how to construct such a subsequence so I began by looking at the $lim inf|a_n|=0$ property although I can't seem to recognize any characteristics about $(a_n)$ except the following.

$lim inf |a_n|=0 \implies \forall\epsilon>0, \exists N'\in\mathbb{N} \space s.t. |inf\{|a_n|:n>N'\}|<\epsilon$

Since $|a_n|\geq0 \space\forall n$,

$0\leq inf\{|a_n|:n>N'\}<\epsilon \implies |a_n|\geq0 \space\forall n>N'$

This is obvious just by observation. I have been having trouble working with $liminf$ and $limsup$ prior to this and was wondering if you guys could offer insight into how to approach these types of problems and what information can be extracted.

Answer 1

Since $\liminf|a_n|=0$, there are infinitely many terms with $|a_n|$ less than any positive number you like. So,

• choose $a_{n_1}$ with $|a_{n_1}|<\frac12$;
• choose a subsequent $a_{n_2}$ with $|a_{n_2}|<\frac14$;
• choose a subsequent $a_{n_3}$ with $|a_{n_3}|<\frac18$;
• and so on.

Then the sum of your chosen terms converges by comparison with the convergent series $\frac12+\frac14+\frac18+\cdots\,$.