# Convergent Subsequence Limit

Prove that:

1. $x_n = 2\sin^3(n) + 6\cos^5(2n)$ has a convergent subsequence.

I understand the concept of a convergent subsequence. If someone could explain to me if there is a proper way of proving this, I would greatly appreciate it.

Thank you

• What kind of sequences do you know that have convergent subsequences? – user21820 Mar 21 '15 at 2:49
• If $\{x_n\}$ is a sequence of reals then by Bolzano-Weierstress theorem it has a convergent subsequence – Empty Mar 21 '15 at 3:37

Since $|x_n|\le 8$ for all $n$, the sequence is contained in a compact subset of $\Bbb R$. Hence, it must have a limit point, and therefore a convergent subsequence.