Bertrand's postulate proof

Bertrand's postulate proof

Regarding
http://michaelnielsen.org/polymath1/index.php?title=Bertrand%27s_postulate
I think the last inequality should be $4^{n/3}\le(2n+1)(2n)^{\sqrt{2n}}$.
But even when the RHS is decreased from $(2n+1)(2n)^{\sqrt{2}n}$, the RHS
still dominates the LHS for $n>>0$ (you can check with wolfram alpha:
http://www.wolframalpha.com/input/?i=%282n%2B1%29%282n%29^{sqrt%282n%29}+-+4^{n%2F3}
).
This doesn't give any contradiction.