Let $g$ be an integrable function such that $g : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ and $g(x) < M$ for all $x$. Let $f_n: \mathbb{Z}^+ \rightarrow \mathbb{R}^+$ such that $$f_n(x) = \sum_{i=0}^{x*n} \dfrac{g\left(\frac{i}{n}\right)}{x}.$$ If $f: \mathbb{Z}^+ \rightarrow \mathbb{R}^+$ such that $$f(x) = \frac{\int_0^x g(t)dt}{x},$$ then $f_n$ is pointwisely convergent to f.
What I am trying to prove is \begin{equation} \lim_{n \rightarrow \infty} \limsup_{x \rightarrow \infty} f_n(x) = \limsup_{x \rightarrow \infty} \lim_{n \rightarrow \infty} f_n(x) = \limsup_{x \rightarrow \infty} f(x). \end{equation}
Since it is not uniformly convergent, I cannot not use Moore-Osgood Theorem. In order to obtain that, I try to use Weierstrass M Test but it is not applicable. Can somebody help me to prove this? Or can you give a counter example? Thank you in advance.