interchanging limits

Question

Suppose I have a double sequence of bounded $L^{p}$ functions $f_{n,m}$ defined on some open set of $\mathbb{R}^{n}$ such that $f_{.,m}$ converges uniformly for each $m$, such that $f_{n,.}$ converges in $L^{p}$ norm for each $n$, and such that $\lim_{n\rightarrow +\infty}\lim_{m\rightarrow +\infty}f_{n,m}$ and $\lim_{m\rightarrow +\infty}\lim_{n\rightarrow +\infty}f_{n,m}$ exists, are these two limits equal ?

Answer 1

My guess is "no".

The details seem like red herrings. The standard counterexample to limit swapping is the sequence of numbers $m/(m+n)$. Doesn't this work? (Ie constant functions)