For the following statements, we must either prove them true or find a counterexample.

1) If $\sum_{n=1}^{\infty}g_n$ converges uniformly, then $(g_n)$ converges uniformly to $0$.

Thoughts: I think this is true and can prove that indeed $(g_n)$ converges to $0$; however, the "uniformly" trips me up and I am unsure how to prove it.

2) If $ 0 \leqslant |f_n(x)| \leqslant g_n(x)$ and $\sum_{n=1}^{\infty}g_n$ converges uniformly, then $\sum_{n=1}^{\infty}f_n$ converges uniformly.

Thoughts: It is simple to see why $\sum_{n=1}^{\infty}f_n$ converges: i.e. comparison test. However, is it suitable to use the Weierstrass M-Test here to prove uniform convergence?

Thanks in advance for any help!

  • 1
    $\begingroup$ The Weierstrass M test won't apply unless the g_n's are just constants. $\endgroup$ – zhw. Apr 12 '16 at 17:44

Hint: In both cases look at the sequence of partial sums, and recall that a sequence of functions converges uniformly iff it is uniformly Cauchy.

  • $\begingroup$ I think I've figured out the first one, so thank you. I am still confused as to the application of this same strategy to the second statement. $\endgroup$ – user322548 Apr 13 '16 at 4:22
  • 1
    $\begingroup$ While Weierstrass M doesn't apply directly, its proof should tell you where to go. $\endgroup$ – zhw. Apr 13 '16 at 4:24

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