Is there a prime number less than the product of consecutive primes, but greater than the last consecutive prime? Let $P(k)$ be the product of $k$ consecutive primes $p_1, p_2, \dots, p_k$. So, e.g. $P(4)$ is $2 \cdot 3 \cdot 5 \cdot 7 = 210$.
Is anything known about whether $P(k) > p_{k+1}$ is always true (for $k > 1$)? It seems like it should be true, since $P(k)$ gets large quickly (faster than $k!$), but I'm having trouble seeing how to construct a counterexample.
If there were a counterexample, then it seems like this would violate PNT locally if it were true since the primes would have to be very sparse in that region. But that's not definitive. Is there an elementary proof one way or another?

By Bertrand's Postulate, which has been a theorem for a long time, there is always a prime between $n$ and $2n$. 
