a question about integral proof: $\lim_{n\to \infty} \int_{0}^\infty {n\cdot {\ln(1+{f(x)\over n}}})dx=\int_{0}^\infty f(x)dx$

Question

A non-negative function ${\rm f}\left(x\right)$ is continuous in $(0,\infty)$ and $\displaystyle{\int_{0}^{\infty}{\rm f}\left(x\right)\,{\rm d}x}$ is convergent.

Then, we need to prove $$\lim_{n\to \infty}\int_{0}^{\infty} n\ \ln\left(1 + {{\rm f}\left(x\right) \over n}\right)\,{\rm d}x =\int_{0}^{\infty}{\rm f}\left(x\right)\,{\rm d}x $$

I think if we can prove $$\lim_{n \to \infty}\int_{0}^{\infty} \left[n\ \ln\left(1 + {{\rm f}\left(x\right) \over n}\right)-f(x)\right]\,{\rm d}x =0 $$, then we can prove it. But I have tried it, I am stuck here. This problem is in my textbook, I don't know how to prove it. Can somebody tell me how to prove it? Thank you!

Answer 1

We have $|n\log(1+\dfrac{f(x)}{n})|\leq |f(x)|$ and $\int_0^1 f(x)dx$ is convergent. Also $n\log(1+\dfrac{f(x)}{n})\to f(x)$. pointwise. Applying the dominated convergence theorem we have the result.