Another algebra problem about prime and induction

[kntsy]

Homework Statement

prove by induction that the $n^{\text th}$ prime is less than $2^{2^{\text n}}$

Homework Equations

hint:assume it is correct for all $n \leq k$, and then compare $p_{k+1}$ with $p_{1}p_{2}...p_{k}+1$

The Attempt at a Solution

is $p_{k+1}$ smaller/greater than $p_{1}p_{2}...p_{k}+1$ so that i can extend the use of inequality?
I attempt to use euclidean algorithm but do not know where to use.
Thank you.

 

[Office_Shredder]

If $$p_{k+1}$$ is larger than the suggested number, you should be able to prove that it has no prime divisors which is a contradiction