Proving $\lim_{n\to+\infty} n\int_{0}^{\alpha}x^{n}\phi(x)dx=0$

Question

Let $\phi : [0,1]\rightarrow\mathbb{R}$ be a Riemann integrable function. How can we show that $\forall\alpha\in[0,1)$, we have : $$ \lim_{n\to+\infty} n\int_{0}^{\alpha}x^{n}\phi(x)dx=0 $$ I need to find an upper bound to be able to find the limit but the problem lies in finding an upper bound for $\phi(x)$. Any hint would be much appreciated, a full answer is not necessary, and thank you!

Answer 1

[Hint] : Notice that $$\displaystyle\left|n\int_{0}^{\alpha}x^{n}\phi(x)\;\text{d}x\right|\leq n\int_{0}^{\alpha}x^{n}\left|\phi(x)\right|\;\text{d}x$$ Now define $\Omega_{x}:=\underset{x\in[0,\alpha]}{\sup}|\phi(x)|$, can you take it from here?