How to show that $n$th prime is less than $2^n$?

Question

I know that there are better lower bounds for number of primes less than n but I don't know how to solve this problem using fundamental proof. I tried using sieve of Eratosthenes and using induction on n to solve the problem but those techniques didn't work. Can someone give me a hint?

Answer 1

The inductive step needs Bertrand's postulate. Every number theorist should read its proof once.

Alternatively, if one gets an upper bound on the relative error in $p_n\sim n\ln n$ (a consequence of the prime number theorem), one can use $n\ln n\in o(2^n)$ to prove the result once a suitable base case is checked.