Let $p$ be a prime number. For every $n \in \mathbb N$, let

$A_{p,n}:=\{\deg P(X) : P(X)\in \mathbb Z[X]$ is monic and $p^n|P(m), \forall m \in \mathbb Z$ $\}$ ; then how to show that $A_{p,n}$ is non-empty ?

If we define $f(n,p):=\min A_{p,n}$, then how to show that $\lim _{n\to \infty}\dfrac {f(n,p)}{n}=p-1$ ?

Let $p$ be a prime number. For every $n \in \mathbb N$, let

$A_{p,n}:=\{\deg P(X) : P(X)\in \mathbb Z[X]$ is monic and $p^n|P(m), \forall m \in \mathbb Z$ $\}$ .

As user abx notes below, $A_{p,n}$ is non-empty.

If we define $f(n,p):=\min A_{p,n}$, then how to show that $\lim _{n\to \infty}\dfrac {f(n,p)}{n}=p-1$ ?