What is an example of a sequence of continuous functions $f_n$ defined on the interval $[0,1]$ where $f_n \rightarrow 0$ pointwise and $\forall n, \int_0^1 f_n(x) dx = 1$?
I've thought of $f_n(x) = nx^n$, but for $x = 1$, $\lim_{n \rightarrow \infty} f_n(1) = +\infty$. I feel like I'm over thinking it and it's something dead simple.