I have a very hard proof from "Proofs from the BOOK". It's the section about Bertrand's postulate, page 9:
It's at the top of the page. We want to know, how often the prime factor p divides $\binom{2n}{n}$. Legendre gives us the solution $\sum_{k \geq 1} \lfloor\frac{2n}{p^k}\rfloor - 2 \lfloor \frac{n}{p^k} \rfloor$ Now the author says, that every addend is max. 1, because we have $\lfloor\frac{2n}{p^k}\rfloor - 2 \lfloor \frac{n}{p^k} \rfloor < \frac{2n}{p^k}-2 \left( \frac{n}{p^k} - 1 \right) = 2$ and the addend is a integral number.
I understood the proof including the sum, but why I leave out the floors in the inequality and why is every addend 1?
Any help is appreciated.