I understood most of the part of the proof except the last 3 lines from "Even though $A<1$, ... $\liminf P_n(0,x)(1-\epsilon)^{-n}>0,...$" part. Mainly I could not understand why occurring this would imply the theorem is true. Please someone give explanation why this is happening. Thanks!
($P_n(0,x)$ in a random walk means the probability of a particle goes from $0$ to $x$ in n-step.)
(Maybe the title of the question is misleading, forgive me, I could not come up with better title.)
[Book: Principles of Random Walk, Author(s): F. Spitzer]
For any particular $\epsilon>0$, $(1-\epsilon)^n$ (the desired lower bound in P2) decays faster than $A(1-\epsilon/2)^n$ (which can be obtained as a lower bound by following the argument given in the proof).