For any prime $p$, is there always at least one prime between $p$ and $2p$?

I conjectured this while thinking about factorials. Obviously, $n!$ is evenly divided by all the whole numbers less than $n$, but what about numbers greater than $n$? I conjectured that the smallest number which does not evenly divide $n!$ is the smallest prime $p \gt n$, and that the smallest composite number which does not evenly divide $n!$ is $2p$. These conjectures are probably false, I haven't proven them, but it did lead me to notice that there tend to be primes between $p$ and $2p$, at least for small values. Is this true for all primes? Or what is the smallest counterexample?

• This is true, and you don't need to specify that $p$ is a prime. It's called Bertrand's postulate...it is not easy to prove. www3.nd.edu/~dgalvin1/pdf/bertrand.pdf – lulu Oct 25 '15 at 2:29
• @lulu It's not difficult either. If you understand elementary number theory, you can understand it, and it's relatively short (2 pages). – user236182 Oct 25 '15 at 4:21