Let $\left(f_n\right)_{n \geqslant 1}$ be a sequence of functions from $\mathbb{N}^*$ to $\mathbb{R}$ such that, for all $x \in \mathbb{N}^*$, the sequence $\left(f_n(x)\right)_{n \geqslant 1}$ converges to a real number $f(x)$ as $n$ tends to $+\infty$. We assume that there exists a function $h : \mathbb{N}^* \rightarrow [0, +\infty[$ such that $h(X)$ has finite expectation and such that $\left|f_n(m)\right| \leqslant h(m)$ for all $m$ and $n$ in $\mathbb{N}^*$. Justify that $E(f(X))$ has finite expectation and show that $$\lim_{n \rightarrow +\infty} E\left(f_n(X)\right) = E(f(X)).$$
Let $\left(f_n\right)_{n \geqslant 1}$ be a sequence of functions from $\mathbb{N}^*$ to $\mathbb{R}$ such that, for all $x \in \mathbb{N}^*$, the sequence $\left(f_n(x)\right)_{n \geqslant 1}$ converges to a real number $f(x)$ as $n$ tends to $+\infty$. We assume that there exists a function $h : \mathbb{N}^* \rightarrow [0, +\infty[$ such that $h(X)$ has finite expectation and such that $\left|f_n(m)\right| \leqslant h(m)$ for all $m$ and $n$ in $\mathbb{N}^*$. Justify that $E(f(X))$ has finite expectation and show that
$$\lim_{n \rightarrow +\infty} E\left(f_n(X)\right) = E(f(X)).$$