We have the following theorem:
If $[a,b]\subset \mathbb R$ and $(f_n)_{n\in \mathbb N}$ is a sequence of integrable functions from $[a,b]$ to $\mathbb R$. Assuming $(f_n)_{n\in \mathbb N}$ converges uniformly to $f$.
Then $f$ is integrable on $[a,b]$ and
$$\int_a^b f=\lim_{n\to\infty} \int_a^b f_n.$$
I know that this theorem is false when we replace $[a,b]$ with $[1,\infty)$.
So I was looking for a counterexample in that particular case, but I didn't find any. Do you think of one ?
Consider the example \begin{align} f_n(x) = \frac{1}{n}\chi_{[1, n+1]} \end{align} which converges uniformly to $0$, but the integrals for the $f_n$ are always $1$ which mean the limit is also $1$ and not $0$.