# Understanding the next proof on a problem of coin toss

I'm reading the next proof in this post.

I have a couple questions about this proof. First, why the inequality $P(A_{k})\leq (2^{k}-k+1)p^{k}$ holds? $A_{k}$ denotes the event of have $k$ or more consecutive heads between the numbered tosses of the problem, so, why is enough such bound only for blocks of $k$ consecutive heads?

And finally, which is the reason of the next: "Hence, the probability of any one of these blocks of $k$ tosses being all heads is $1-(1-p^k)^{\left\lfloor\tfrac{2^k}{k}\right\rfloor}.$"

Any kind of help is thanked in advanced.