Product of all primes less then x

Question

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

Answer 1

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.