How to prove the existence of a prime number with elementary method?

Question

Suppose $A>5$ is an integer and there exists a prime number $p$ such that $A-2\leq p^2<A$.

Show that there exists at least one prime number $q$ such that $p<q<A$.

This seems to be intuitive to me since $A$ and $p$ has large gap. How can we prove it?

Answer 1

Using one version of Bertrand's Postulate, if $q$ is the next prime after $p$ then $$p<q<2p\le p^2<A\ .$$