I'm trying to show:
If $f \in L^1([0,1])$, then $$ \lim_{n \to \infty}\int_0^1n \ln\left(1+\frac{|f(x)|^2}{n^2} \right)dx=0 $$
Solution: I want to apply de Dominated Convergence Theorem. However, I don't find a suitable $h \in L^{1}([0,1])$ such that $$ \left| n \ln\left(1+\frac{|f(x)|^2}{n^2} \right) \right|=\left|\ln\left( \frac{n^2+|f(x)|^2}{n^2}\right)^n \right| \leq h(x) \quad \quad \forall n \in \mathbb{N}. $$ Any hint will be appreciated.
Thanks.