Regular ring/Characterication with Hilbert-Kunz multiplicity/Fact

Let ${}R$ be a local noetherian ring of positive characteristic. Then the following hold.

1. The Hilbert-Kunz multiplicity of ${}R$ is ${}e_{HK}(R)\geq 1$.
2. If ${}R$ is unmixed, then ${}e_{HK}(R)=1$ if and only if ${}R$ is regular.

Write a proof