In the book of Writing Proofs in Analysis by J.M Kane, at page 78, it is given that
However, in the case 2, he states that since $p$ is an accumulation point of $A$, there exists $k> N$, but the fact that $p$ is an accumulation point only says that $(B(p,\epsilon)- \{ p\}) \cap A \not = \emptyset$, hence there exits $n_1 \in \mathbb{N}$ s.t $x_{n_1} \in B(p,\epsilon)$, so how can the author conclude from that $n_1 > N$ ?
You are missing a quantifier: the fact that $p$ is an accumulation point says that for every positive radius $r$ the intersection $(B(p,r)- \{ p\}) \cap A$ is nonempty. This in turn implies that the intersection is infinite, for all radii. (Why? Consider what happens for $r=1/n$. We shrink the ball each time we increment $n$, yet still the intersection is nonempty!)
Choosing the radius to be $\epsilon/2$, we see there are infinitely many $a_n$ within $\epsilon/2$ of $p$. Hence there must be a $k>N$ such that $a_k$ is within $\epsilon/2$ of $p$.