Bounded functions converge uniformly

[Imum Coeli]

Q: Let \(\displaystyle D \subseteq \mathbb{R}\) and \(\displaystyle f_ n : D \to \mathbb{R}\) be a function with \(\displaystyle |f_n(x)|\leq M_n \) for each \(\displaystyle n \in \mathbb{N}\). Show the series \(\displaystyle \sum_{n=1}^{\infty} f_n\) converges uniformly on \(\displaystyle D \) if \(\displaystyle \sum_{n=1}^{\infty} M_n < \infty\).

Again I am having trouble starting. I thought that I could use the fact that if \(\displaystyle (f_n) \) is uniformly Cauchy on D, then there is a function \(\displaystyle f : D \to \mathbb{R} \) such that \(\displaystyle (f_n) \) convereges uniformly to \(\displaystyle f \) on D. However I don't see how the fact \(\displaystyle \sum_{n=1}^{\infty} M_n < \infty\) gives me anything useful to work with.

The best I can come up with so far is the inequality \(\displaystyle |f_n(x) -f_m(x)| \leq M_n - M_m\) but that's not very helpful...

 

[Imum Coeli]

Okay...

\(\displaystyle \sum_{n=1}^{\infty} M_n<\infty \implies \;\forall\; k \in \mathbb{N} \exists B \in \mathbb{R} : \sum_{n=1}^{k} M_n \leq B \) (does not depend on x)

Since \(\displaystyle \sum_{n=1}^{\infty} M_n\) is non-negative termed and the sequence of partial sums \(\displaystyle \sum_{n=1}^{k} M_n\) is bounded, \(\displaystyle \sum_{n=1}^{\infty} M_n\) converges to some \(\displaystyle L \in \mathbb{R} \).

Therefore \(\displaystyle \forall \epsilon > 0 \exists N \in \mathbb{N} : n>N \implies |M_n-L|<\epsilon \)

Now \(\displaystyle |f_n(x)| \leq M_n \iff -M_n \leq f_n(x) \leq M_n \iff -(M_n+L) \leq f_n(x)-L \leq M_n-L \)

Thus \(\displaystyle |f_n(x)-L| \leq |M_n-L| < \epsilon \implies |f_n(x)-L| < \epsilon \;\forall\; x\in D \) if \(\displaystyle n > N\)

Therefore \(\displaystyle \sum_{n=1}^{\infty} f_n(x) \) converges uniformly on D.