Are There Proofs of Bertrand's Postulate that Don't Use Case Work?

Question

There are multiple fairly pleasant, reasonably transparent proofs of Bertrand's Postulate, that for all integers $n>0$ there is always a prime (weakly) between $n$ and $2n$. All of the ones I'm aware of somewhere along the line check a small number of exceptional cases directly. This is unpleasant. The Prime Number Theorem gives us that this holds asymptotically for $2$ replaced by any positive number $a$, which gives that it must hold for all $n$ for some $b$, and obviously $2$ is the smallest possible value this $b$ could take on. So it feels like there should be a "reason" for $b$ to take on the very specific value of $2$, as opposed to the result holding asymptotically for $2$, and then holding in all the remaining cases by "chance". So, are there any proofs of Bertrand's Postulate which avoid case work entirely?