In my complex analysis class I was grading a problem of the following type:
given where is compact and given a sequence of continuous functions which converges to 0 pointwise on and if for all show that the convergence is uniform.
In a previous post, I talked about why it was important that each be continuous.
Now what about the hypothesis? Can it be dispensed with?
Answer: well, no.
Let’s look at an example in real variables:
Let with . for all . To see that converges to zero pointwise, note that for all , hence which implies that by the squeeze theorem. But does not converge to 0 uniformly as for we have