# Confusion: Proof of Bertrand's Postulate, Primorial function upper bound

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!

• I have explained Bertrand's postulate here, you can check if it helps. – tarit goswami Feb 1 at 12:23
• Thanks so much. How do you make this assertion: $$\prod_{p\le 2n-1} p< 4^{2n-1}$$ – jonan Feb 1 at 14:46
• 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 $n\ge 507$.. – tarit goswami Feb 1 at 17:55