Help understanding real analysis proof?

Posted on 2016-02-22 19:11:00 in Math by ALEX

The problem is to prove the limit of x^(1/3) as x approaches c is c^(1/3), c>0, x>0. The proof is: "for any sequence a_n such that lim as n approaches infinity of a_n=c, we want to show limit as n approaches infinity of a_n^(1/3)=c^(1/3). By monotonicity of f(x)=x^(1/3), (a_n)^(1/3) is bounded." I'm not sure how the sequence being bounded follows from the function being monotone. 

"By the Bolzano-Weierstrass theorem, there exists a convergent subsequence (a_n_k)^(1/3). Its limit is L=c^(1/3)." Why is its limit c^(1/3)? I feel like that just comes out of nowhere. 

"Recall a sequence is convergent to L iff for any subsequence of it, there exists a further subsequence which converges to L. So we can conclude." I think I understand what this means, but what does it have to do with what we did in the proof before? 

Thanks in advance for any help.

---

Answer Comment

No Answers Posted Yet.