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?

  • $\begingroup$ I´m not sure but I think its key consider A open or compact $\endgroup$
    – JoseSquare
    Feb 12 '19 at 22:32
  • $\begingroup$ I don't know what that means. $\endgroup$ Feb 12 '19 at 22:35
  • $\begingroup$ 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 $\endgroup$
    – JoseSquare
    Feb 12 '19 at 22:38
  • 1
    $\begingroup$ See the comment on top answer of math.stackexchange.com/questions/26273/… for a counterexample. $\endgroup$
    – cofnmarol
    Feb 12 '19 at 23:09
  • $\begingroup$ But what's the meaning of compact set and open set? $\endgroup$ Feb 12 '19 at 23:38

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