# True/False: Real Analysis: Series of Functions

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!

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