$|s_{2k}-s_k|<\epsilon$ implies $\{s_k\}$ converges? $\{s_k\}$ is a sequence in $\mathbb{R}$.
$\forall\epsilon>0$, $\exists N\in\mathbb{N}$ s.t. $k\geq N$ implies $|s_{2k}-s_k|<\epsilon$.Then, does $\{s_k\}$ converge or not?
I need a proof. Thanks.

No, it does not. Define your sequence as you wish for odd numbers $k$ and 
extend $s_k=s_{2^lk}.$ 
