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

Question

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?