The problem is from Chapter 8 of the book, and it states the following.

"In a sequence of Bernoulli trials let $A\_n$ be the event that a run of $n$ consecuitive successes ocurrs between the $2^n$th trial and the $2^{n+1}$st trial. If $p \ge \frac{1}{2}$, there is probability one that infinitely many $A\_n$ occur; if $p<\frac{1}{2}$, then with probability one, only finitely many $A\_n$ ocurr"

I've had several issues when trying to solve this problem, but the most important one is how the book solves this problem:

"Note that $Pr(A\_n) < (2p)^n$, but $Pr(A\_n) > 1 - (1-p^n)^{\frac{2^n}{2n}} > 1-e^{\frac{(-2p)^n}{2n}}$.

If $p=\frac{1}{2}$ the last quantity is approximately $\frac{n}{2}$; if $p>1$ then $Pr(A\_n)$ does not even tend to zero".

Can someone explain the proof that is given? Because it kinda confused me.