Erdos's proof of Bertrand's postulate

In 1845 Bertrand postulated that there is always a prime between $n$ and $2n$, and he verified this for $n < 3,000,000$. Tchebychev gave an analytic proof of the postulate in 1850. In 1932, in his first paper, Erdos gave a beautiful elementary proof using nothing more than a few easily verified facts about the middle binomial coefficient. We describe Erdos's proof and make a few additional comments, including a discussion of how the two main lemmas used in the proof very quickly give an approximate prime number theorem. We also describe a result of Greenfield and Greenfield that links Bertrand's postulate to the statement that $\{1,\ldots,2n\}$ can always be decomposed into $n$ pairs such that the sum of each pair is a prime.