Limit point versus limit and implications for convergence
Solution 1:
The sequence $2,1/2,3,1/3,4,1/4 \cdots, n,1/n \cdots$ has the only limit point $0$, but not converges.
Solution 2:
You could also pick the sequence
$ (1 + (-1)^n).n $
which doesn't converge but has 0 as limit point.