http://mathworld.wolfram.com/ChoquetTheory.html

Is the claim in the link true? Here's the reference given there:

https://www.renyi.hu/~p_erdos/1934-01.pdf

Erdős proved that there exist at least one prime $\equiv 1\pmod{4}$ and at least one prime $\equiv 3\pmod{4}$ between $n$ and $2n$ for all $n>6$.

References:

Erdős, P. "A Theorem of Sylvester and Schur." J. London Math. Soc. 9, 282-288, 1934.

Also: https://en.wikipedia.org/wiki/Talk:Bertrand%27s_postulate#Dubious_statement

My question:

Can you find the proof of this in the given pdf (maybe somewhere else)?

EDIT: For those interested, there's a generalization of this fact:

https://books.google.lt/books?id=tmORL-UYOyEC&pg=PA386&lpg=PA386&redir_esc=y#v=onepage&q&f=false