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In my textbook, the problem is as follows:

{$x_n$} is a convergent sequence and k is in N. Prove $\lim_{n\to\infty}$ $x^k_n$ = ($\lim_{n\to\infty}$ $x_n$)$^k$ with the use of induction.

So normally with induction you would have a base case and show that it holds. Then right after, you prove that for every n+1 it will hold as well and thereby finishing the proof. However, I'm not sure how I would do that with this problem. Any help would be appreciated.

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    Please use MathJax (http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) for formatting your formulas. As they are written now it's hard to see what you want to prove. – 2017-02-23
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    @skyking Does my edit make it clearer now? I apologize, I'm not that familiar with formatting on here. – 2017-02-23
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    Write $$\lim_{n\to\infty}x_n=L$$ and assume that $$\lim_{n\to\infty} x^k_n=L^k.$$ Then $$\lim_{n\to\infty} x^{k+1}_n=\lim_{n\to\infty} x^k_n\cdot x_n=L^k\cdot L=L^{k+1}.$$ – 2017-02-23
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    Do you have to use induction? I think it would be quite straight forward to prove without. One could of course throw in induction somewhere in the proof just for the sake, but I think that's kind-of weird. – 2017-02-23
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    @skyking Induction is a requirement for the problem. – 2017-02-23

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The case where $k=1$ is trivial. Note that $\lim_{n}a_{n}b_{n} = (\lim_{n}a_{n})(\lim_{n}b_{n}$). Suppose there is some $k \geq 1$ such that the statement pertaining to $k$ is true. Then $$ \lim_{n}x_{n}^{k+1} = \lim_{n}x_{n}^{k}x_{n} = (\lim_{n}x_{n}^{k})(\lim_{n}x_{n}) = (\lim_{n}x_{n})^{k}(\lim_{n}x_{n}) = (\lim_{n}x_{n})^{k+1}. $$