My question:
In the proof below for the strong LLNs, I don't know how the author goes from the $\color{red}{\text{red box}}$ to the $\color{blue}{\text{blue box}}$ (see screenshot below)
Getting to the red box makes sense to me. The red box says that each term in the blue box is finite. Note that the very beginning of the proof states $K = E[X_i^4] < \infty$
But just like $1+1/2+1/3+...$ goes to infinity, how does showing every term of the sequence is finite prove that the series converges?
I forgot all this stuff from HS Calculus and could use help, thank you.
Proof from the book:
