Bertrand's Postulate asserts that there is a prime between $n$ and $2n$.
Is this the best such upper bound on prime gaps known today, or have stronger estimates been proved? I mean results of the kind:
- there will always be a prime between $n$ and $2n-2$, or
- there will always be a prime between $n$ and $cn$ with $1<c<2$?
Was any such improvement proved rigorously, or is Bertrand's Postulate still the best we have?