If $f \in L(0,1)$ show that $x^{n} f(x) \in L(0,1)$ for $n \in \mathbb{N}$ and that $\displaystyle \int_{0}^{1} x^{n} f(x) dx \rightarrow 0$ as $n \to \infty$.
Is the following attempt correct?
Since $0 Now to finish the second part can we simply say that: $x^{n} f(x) \rightarrow 0$ as $n \rightarrow \infty$ pointwise so we may apply convergence theorem.