This is a counter example homework question that I can't seem to solve.
I need to find a sequence of real numbers $(x_n)_{n=1}^{\infty}$, and a monotonic increasing sequence of natural numbers $(n_k)_{k=1}^{\infty}$such that: $(i)\lim\limits_{n\to\infty}(x_n-x_{n+1})=0$, $(ii)(x_{n_k})_{k=1}^{\infty}$ converges, but $(iii)(x_n)_{n=1}^{\infty}$ do not converges.
Only thing I know so far is that for all $k$, $[n_{k+1}-n_k]$ cannot be bounded.