Existence of a Brownian motion - Proof.

Question

I am reading a proof by Lalley of the existence of a Brownian motion. In this final part, he argues, that the geometric series converges a.s.

(see image; or see link: http://galton.uchicago.edu/~lalley/Courses/385/BrownianMotion.pdf)

I have a very hard time following this argument.

Firstly, I dont understand how to justify that

$$\max_m\lvert X_{n,m}\rvert\leq2^{n/4}\quad \forall\quad n>N$$

is neccesary / sufficient to ensure that geometric converges.

More importantly, I dont understand how the Borel-Cantelli lemma and the crude union bound finish the job.

Can anyone help me clarify? Some equations showing the steps would be very helpful! Or just a more in-dept explanation of what is going on here.

Thanks in advance!