I have a sequence $(a_n)$. Its subsequence $a_{2^n}$ converges to a finite limit $g$. The sequence $|a_{n+1} - a_n|$ converges to $0$. Can we conclude about the convergence of $(a_n)$? I suspect I have to notice this is a Cauchy sequence, however, I'm not sure how to use the fact that $a_{2^n}$ converges to $g$.
The Cauchy condition for convergence states that $(a_n)$ converges if and only if $$\forall \epsilon>0 , \exists n_{\epsilon} \in \mathbb{N} , s.t.\forall m,k > n_{\epsilon}, |a_m - a_k|<\epsilon.$$ Alternatively, we could show that $(a_n)$ is bounded from above and monotonic, however, I can't quite arrive at this conclusion either.