I was one of those who voted to close -- the problem seemed unmotivated and simply presented as something seen on AoPS. Here are some suggestions on perhaps reformulating the question, and on the math itself. If you'd like to edit your question, I'm happy to give the benefit of doubt and will vote to reopen. But perhaps what I say here will be enough for you.
The question is better formulated as "Is there a prime in the interval $[n,n+\pi(n)]$ for every $n\ge 2$?" Note that this is a good deal stronger than Bertrand's postulate asking for a prime in $[n,2n]$, which anyway has a simple elementary proof. Your question follows easily for large $n$ by the prime number theorem with a strong error term; possibly we know enough to check it for all $n$. Since the prime number theorem with a strong error term has an elementary proof (in the sense of avoiding complex analysis), so does your question. But perhaps what you want is a simple proof; this may not be easy.
Anyway the problem as stated seemed unsuitable to me, and even with the reformulation I think it is borderline, but as I said I'd be happy to give the benefit of doubt.