Question about a theorem of david williams:probability with martingales

Question

I am reading David Williams: Probability with martingales and got a questions:

At first a theorem:

Now I want to know why the following holds:

The full proof of b):

I was reading this prove many times. But unfortunately I can't see how the statement follows by the proof of b). Of course I tried it by myself, but (a) just yields that $\sum Var(X_k)=\infty$ if $\sum X_k=\sum \mathbb E(X_k^2)=\infty$, but I don't know how to connect this with the partial sums of $X_k$

I really hope that someone can go through it.

Any help is much appreciated.