Has this partial result about Legendre's Conjecture been proved?

Question

I'd like to know if it has been proved this "partial result" about the Legendre's Conjecture:

(1) There are infinitely many $n$ such that there's a prime in $(n^2, (n+1)^2)$

Thanks!

Answer 1

Since a prime number is never a square the set of all prime numbers is contained in $\cup_{n \in \mathbb{N}} (n^2, (n+1)^2)$, which are all integers but the squares. Since there are infinitely many primes while each finite collection of such intervalls is finite the results follows.