A counterexample to "the integral of the limit is the limit of the integral"

I was going to append this to @chwala 's thread here , but thought it deserved a new thread. For ##n \geq 1##, define $$f_n : [0,1] \to \mathbb{R} : x \mapsto \begin{cases} 1 & x = 0, \\ n(1 - nx) & x \in (0, \tfrac 1n], \\ 0 & x \in (\tfrac 1n, 1]. \end{cases}$$ (Note that ##f_n## is... Read more