I am very unsure so maybe somebody else can answer too, but I think it is about order of taking limits. I thought true at first too but I remember sometimes the order of limits can give different result. If the order of limits did not mater then you can change to take first the limit as k goes to infinity of the finite integral from 0 to A of f_k. This will be 0 for any A because f_k ---> 0 uniformly and it is ok to switch order of limit and integral when there is uniform convergence of sequence of functions. Then taking other limit as A goes to infinity will still be 0. You must find example that order of taking limits is different. I am sorry I can not make one I have been away from math many years that is why my name is memory loss LOL