If a sequence $\{S_n\}$ converges to a constant $s$, then the sequence $\{S_n^k\}$ converges to $s^k$ Let $\{s_n\}$ be a sequence in $\mathbb{R}$, and assume that $s_n \rightarrow s$. Prove that $s^k_n\rightarrow s^k$ for every $k \in\mathbb{N}$
Ok, so we need $|s^k_n - s^k| < \varepsilon$. I rewrote this as
$$|s_ns^{k-1}_n - ss^{k-1}|=|(s_n-s)(s^{k-1}_n + s^{k-1}) -s_ns^{k-1}+ss_n^{k-1}|$$
But this seems really messy. What should I use here: $|s_n - s| < \varepsilon?$
Help!

It suffices to show that the function $f(x) = x^k$ is continuous.  Now clearly the function $g(x) = x$ is continuous ($\delta = \epsilon$).  Now just prove that the product of continuous functions is continuous and then use induction.
