Prove there is a subsequence $(a_{nk})_{n=1}^\infty$ such that $\Sigma^{\infty}_{k=1} a_{nk}$ converges.

Question

Hey everyone this was give as a practice problem and i'm having trouble, any help is appreciated

Let $(a_n)_{n=1}^\infty$ be a sequence such that $\displaystyle \lim_{n \rightarrow \infty} {a_n} = 0$. Prove there is a subsequence $(a_{nk})_{n=1}^\infty$ such that $\Sigma^{\infty}_{k=1} a_{nk}$ converges.

Answer 1

Using the fact that $\displaystyle \lim_{n \rightarrow \infty} {a_n} = 0$, you know that for every $k$, there exists such an $n$ that $|a_n| < \frac{1}{2^k}$. Now construct the sequence like so:

1. $n_1$ is the first value of $n$ for which $|a_n|<\frac12$.
2. $n_k$ is the first value of $n>n_{k-1}$ for which $|a_n|<\frac{1}{2^k}$