Originally Posted by

**Moo** Hello,

Firstly, you have to prove that $\displaystyle \lim_{n \to \infty} \int_0^1 g_n(x) ~dx=\int_0^1 \lim_{n \to \infty} g_n(x) ~dx$, by using the dominated convergence theorem for example (f is continuous over a closed set, --> it is bounded)

Secondly, you have to prove that $\displaystyle g_n(x) \xrightarrow[n \to \infty]{} f(0)$

In order to show that, write the epsilon definition of this limit : $\displaystyle x^n \xrightarrow[n \to \infty]{} 0$, when $\displaystyle x \in [0,1)$ (you can "miss" the case x=1, because it's a neglectible case when dealing with integrals... dunno if you know how to deal with it)

Then use the epsilon-delta method on the continuity of f, and you're done.