I believe there was an old conjecture that there's **always a prime number between `N` and `2N`**.


What's the history and how is this proven is the easiest/elementary/deepest ways?



Bertrand's postulate
Proven. This is called Bertrand's postulate. Here is [Erdos' elementary proof](http://www.nd.edu/~dgalvin1/pdf/bertrand.pdf); the original proof is due to Chebyshev.


