An example of a sequence which does not have any subsequence with a finite limit. 
Find and example of a sequence $\{x_n\}$ which does not contain any subsequences having a finite limit.

I've been thinking of the following seqeunce:
$$
x_n = \sin(n)\cdot\sin(\sqrt{3}n)
$$
But is that true? Also what are other examples of such sequences?

The example in the OP is wrong. This can be shown by Bolzano-Weierstrass theorem which states that every bounded sequence contains a convergent subsequence.
Now if we consider:
$$
x_n = \sin(n)\sin(\sqrt{3}n)
$$
its clear that the sequence is bounded by $[-1, 1]$. Hence it contains a convergent subsequence.
