The proofs for $(2n,3n)$ and $(3n,4n)$ are elementary and very pleasing (based on a quick look.) It is known that there is always a prime between $k$ and $\frac{6k}{5}\ $ for $k \gt 24.$ The proof is more involved but do not use anything analytic. Putting $k=2n$ etc gives a prime in $(4n,\frac{24n}{5})$ (except $n=1,2,6$) and hence in $(4n,5n)$ except for $n=1,2$ and one in $(5n,6n)$ except for $n=1$.

The proofs for $(2n,3n)$ and $(3n,4n)$ are elementary and very pleasing (based on a quick look.) It is known that there is always a prime between $k$ and $\frac{6k}{5}\ $ for $k \gt 24.$ The proof is more involved but do not use anything analytic. Putting $k=2n$ etc gives a prime in $(4n,\frac{24n}{5})$ (except $n=1,2,6$) and hence in $(4n,5n)$ except for $n=1,2$ and one in $(5n,6n)$ except for $n=1$.