Problem Statement: Let and let for all and for every . Show that if converges then converges uniformly on .
Note: This is asking us to prove the Weierstrass M-Test.
Proof: Let and for every and for every . Assume that converges. This implies that the sequence of partial sums, , converges. Every convergent sequence is Cauchy.
We wish to show that for every there exists such that for all and for every it follows that . Note that .
Let . Then there exists a such that for it follows that . This implies that for .
Let and without loss of generality suppose that . Then it follows that:
Thus, converges uniformly.
Reflection: I’m really glad I did this proof. This has helped me remember the Weierstrass M-Test so I can use it on the qual if I need to, but it’s also a really good technique to know. When trying to show something converges uniformly you just need to show that the sequence of partial sums is Cauchy for every in your domain.