This question's my profesors. I know $x^n$ is a pointwise convergent sequence. But I don't know how can I show that this limit. Can you show me solution of this question.
$f:[0, 1]\rightarrow \mathbb{R}$ continuous function.
$$\lim_{n\rightarrow \infty }\int_0^{1}x^nf(x) \, dx$$
Find the limit.
Since $f$ is continuous, it is bounded on $[0,1]$, say by $M >0$. Then,
$$\left\lvert \int_0^1 x^n f(x) \,dx \right\rvert \leq \int_0^1 x^n |f(x)| \,dx \leq M \int_0^1 x^n \,dx = \frac{M}{n+1} \to 0$$
as $n \to \infty$.