A true/false question from a quiz a while back. It's still bugging me because I'm told it's a false statement, but after three weeks I can't think of a counterexample. Maybe I just haven't quite made sense of it. A suitable counterexample or even a clarifying explanation of what this is saying would be of great help. Thanks in advance--here's the statement:

If $\displaystyle \int _0 ^{\infty} f_k(x) dx = \lim _{A\to \infty} \int _0 ^A f_k(x) dx $ exists for each $\displaystyle k \in \mathbb{N} $ and $\displaystyle f_k \to 0 $ uniformly on $\displaystyle [0,\infty ) $, then $\displaystyle \lim _{k\to \infty } \left(\lim _{A\to \infty } \int _0 ^A f_k(x) dx \right) = 0 $.