Question: Prove that if $x_n$ is a convergent sequence, $k \in N$, then $\lim (x_n^k) = (\lim x_n)^k$. Hint: use induction.
Let $\lim x_n = x.$ Then, $(\lim x_n)^k = x^k$. So, I have to show that $\lim (x_n^k)=x^k$.
Do I have to show this $|x_n^k-x^k| < \varepsilon$ by using induction?? I can't proceed from here. Could you give some hint??
Thank you in advance.
It concerns induction on $k$.
Prove (or apply if it is at your disposal allready) that: $$[a_n\to a\text{ and }b_n\to b]\implies[a_nb_n\to ab]$$
Then the induction step will be
$$[x_n^{k-1}\to x^{k-1}\text{ and } x_n\to x]\implies[x_n^k=x_n^{k-1}x_n\to x^{k-1}x=x^k]$$