0
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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
2
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.