# Reciprocal of the Weierstrass $M$ test.

I need help with this problem:

Suppose that $$f_n$$ are continous non-negative fucntions bounded on A and let $$M_n=\sup f_n$$. If $$\sum_{n=1}^\infty f_n$$ converges uniformly on A, it follows that $$\sum_{n=1}^\infty M_n$$ converges? (a reciprocal of the Weierstrass $$M$$ test

I don't know how to solve this problem. Can you give me some help please?

• I´m not sure but I think its key consider A open or compact Feb 12 '19 at 22:32
• I don't know what that means. Feb 12 '19 at 22:35
• My intuition tells me that if A is a compact set then the theorem might be true, but if A is an open set then I think a counterexample might be found. This is only intuition, nothing serious Feb 12 '19 at 22:38
• See the comment on top answer of math.stackexchange.com/questions/26273/… for a counterexample. Feb 12 '19 at 23:09
• But what's the meaning of compact set and open set? Feb 12 '19 at 23:38