$(1+p)^{p^{n-1}}\equiv1\pmod{p^n}$ is true for $n=1$,

Suppose the equivalence is true for some $n\ge1$, then
$$
\begin{align}
(1+p)^{p^n}
&=\left((1+p)^{p^{n-1}}\right)^p\tag1\\[12pt]
&=(1+cp^n)^p\tag2\\[9pt]
&=1+p\left(cp^n\right)+\sum_{k=2}^p\binom{p}{k}\left(cp^n\right)^k\tag3\\
&=1+p^{n+1}\left(c+\sum_{k=2}^p\binom{p}{k}c^kp^{n(k-1)-1}\right)\tag4
\end{align}
$$
Explanation:
$(1)$: $x^{ab}=\left(x^a\right)^b$
$(2)$: inductive hypothesis (the equivalence is true for $n$)
$(3)$: the Binomial Theorem