Prove that $\mathbb{Z}_p$ is isomorphic to $End(\mathbb{Z}_p)$

Question

Here $\mathbb{Z}_p$ is p-adic integer. The isomorphism $End(\mathbb{Z}_p) \cong \mathbb{Z}_p $ appears on the Silverman's textbook "elliptic curve". It seems it can be proved using an action on $\mathbb{Z}/p^r \mathbb{Z}$ induced by endomorphism on $p^r\mathbb{Z}$ to itself. But I am not sure about details. So how to prove this?