My number theory assignment walks me through the proof of Bertrand's postulate. The steps taken are essentially the same as the ones shown here: https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate
I have worked out everything until the induction bit. The part I am stuck on is as follows:
I think I have proved the divisibility bit. I have gone over the proof I linked above, but I still seem to lack an understanding of the final argument made. I showed that:
$$\frac{\left(2m-1\right)\#}{m\#}\hspace{3pt} | \hspace{3pt} \binom{2m-1}{m}$$
And I also know that:
$$\binom{2m-1}{m} \leq 2^{2m-1}$$
However, I do not know how to proceed with this argument. Any information/advice would be greatly appreciated. Thanks!