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How to prove that $\prod_{p\leqslant{x}}p\leqslant4^{x-1},\ \forall x\geqslant2$,
where product is taken over all prime numbers $p\leqslant{x}$

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this MO link could help, – Raymond Manzoni Aug 10 '12 at 18:46
up vote 5 down vote accepted

One proof of this fact is the proof of Claim 2.2 of these notes on Erdos' proof of Bertrand's postulate by David Galvin.

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