For every $n \in \mathbb {N} $, assume $p_{n}$ is the $n^{th}$ prime number. if $n \gt 1$, prove that $p_{n}\lt 2^{n}$.

Question

I tried to solve this problem using mathematical induction. First, we say that $ 3 \lt 4$ for $n =2$, which is correct. Then by assuming that it's accurate for $k$, we need to prove it's accurate as well for each $ k+1$. $$p_{k+1} \lt 2^{k+1} \Rightarrow p_{k+1} \lt 2^{k} \times 2 \Rightarrow \cfrac{p_{k+1}}{2} \lt 2^{k} $$ I don't know how should I prove it at last and I don't know if this is the way of solving this or not. Can someone correct me if I'm wrong and help me figure out the rest?