Example of $a_{n}$ such that $\lim_{n\to\infty }a_{n}=0$, and there is a convg subsequence $\left ( S_{k_{n}} \right )$ of $\left ( S_{n}\right )$ I'm having trouble to find an example of a sequence $\left ( a_{n} \right )$ such that $\lim_{n\to\infty }a_{n}=0$, and there is a convergent subsequence $\left ( S_{k_{n}} \right )$ of $\left (  S_{n}\right )$, but $\sum a_{n}$ diverges.
We denote $S_{k}=a_{1}+a_{2}+\cdot \cdot \cdot +a_{k}$
My first approach was finding $\left (  S_{n}\right )$ that is bounded but divergent (because this will lead us to conclude that there is a convergent subsequence). For example, there is $S_{n}=sin(n)$, but then $\lim_{n\to\infty }a_{n}\neq 0$. So, it fails to be our example.
Is there a better way to find such examples?

Hint:
Try to construct $a_n$ such that $S_n$ oscilates from $1$ to $-1$ and back to $1$ and back to $-1$. Each oscilation will have to take more steps, since $a_n$ will have to be smaller and smaller, but you can afford it, you can have $a_n$ stuck on a particular value for how ever long you want.
