Hello, I am trying to prove that given a proper integral, , assuming both converge.
Here's my attempted proof:
, where as .
Thus .
But as .
Hence .
Does this look valid?
There's a mistake in this step, since we don't know that we can bound uniformly on the interval.
That said what you're trying to prove is false as stated (indeed, if it were true, theorems like dominated and monotone convergence would be unecessary), there is a sequence of functions such that all terms have the same integral and the sequence converges pointwise to zero.