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

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

  • $\begingroup$ In a nutshell: compactness is equivalent to sequential compactness in metric spaces. $\endgroup$ – Math1000 Mar 21 '15 at 3:11

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