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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](http://mathoverflow.net/questions/10496/an-inequality-relating-the-factorial-to-the-primorial) could help,2012-08-10

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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.