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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:

enter image description here

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!

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    $\begingroup$ I have explained Bertrand's postulate here, you can check if it helps. $\endgroup$ – tarit goswami Feb 1 at 12:23
  • $\begingroup$ Thanks so much. How do you make this assertion: $$ \prod_{p\le 2n-1} p< 4^{2n-1}$$ $\endgroup$ – jonan Feb 1 at 14:46
  • $\begingroup$ While proving the previous lemma we need some inequality like the one you mentioned. $\prod_{p\le 2n-1}p<4^{2n-1}$ is the simplest choice.. and it's comes out to be true for $n\ge 507$.. $\endgroup$ – tarit goswami Feb 1 at 17:55

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