I have a very hard proof from "Proofs from the BOOK". It's the section about Bertrand's postulate, page 8:
It's about the part, where the author says:
$\binom{2m+1}{m}\leq 2^{2m}$ because $\binom{2m+1}{m}=\binom{2m+1}{m+1}$ are the same in $\sum \limits_{k=0}^{2m+1} \binom{2m+1}{k}=2^{2m+1}$
I see, why they are the same, but I don't see the reason to say $\binom{2m+1}{m}\leq 2^{2m}$. Any help would be fine :)