Suppose $\{f_n\}, n\in \mathbb{N}$ are functions from $[a,b]\to \mathbb{R}$, Riemann integrable on $[a,b]$, and converge pointwise to $f$, which is also Riemann integrable on $[a,b]$. Also, $\int_a^b f_n(x) dx = 1$ for all $n$. Prove or give a counterexample that $\int_a^b f(x) dx \leq 1$.
I think the result is true but can’t prove it.
Try $f_{n}(0)=2$, $f_{n}(x)=\dfrac{2}{1-1/n}$ for $1/n\leq x\leq 1$, $f_{n}(x)=-n$ for $0<x<1/n$, then the pointwise limit is $f(x)=2$.