J. Nagura, already in 1952, proved that the interval $(x,\frac{6}{5}x)$ contains a prime for any $x \ge 25$. The proof appears in the paper "On the interval containing at least one prime number" published in Proc. Japan Acad. Volume 28, Number 4 (1952), 177-181 (see this link). From Nagura's theorem one obtains the results mentioned in your post by choosing $x=2n$ or $x=3n$, and checking what happens for small $n$.

EDIT: Jose Brox, in another answer, had provided references which not noly predate Nagura's result, but they are also stronger.

J. Nagura, already in 1952, proved that the interval $(x,\frac{6}{5}x)$ contains a prime for any $x \ge 25$. The proof appears in the paper "On the interval containing at least one prime number" published in Proc. Japan Acad. Volume 28, Number 4 (1952), 177-181 (see this link). From Nagura's theorem one obtains the results mentioned in your post by choosing $x=2n$ or $x=3n$, and checking what happens for small $n$.