This is not a hard question, but I am stuck trying to prove the following by induction: Let $$\begin{align}C_1&\equiv C^e\pmod n\text{ and}\\ C_{j+1}&\equiv C_j^e\pmod n\end{align}$$ for all $j\in\mathbb{N}$, with $0 < C_1 < n$ and $(e,\phi(n))=1$.
Show that $C_j\equiv C^{e^j}\pmod n$, with $0 < C_j < n$.
This is what I have tried:
Base:
$$C_1\equiv C^{e^1}\pmod n.$$
($j=1$):
$$C_2\equiv C_1^e\equiv(C^e)^e\equiv C^{e^2}\pmod n.$$
Hence the base case is met.
Step:
$$C_{k+1}\equiv C_k^e\pmod n.$$
($j=k+1$):
$$C_{k+2}\equiv C_{k+1}^e\equiv(C_k^e)^e\equiv C_k^{e^2}\pmod n.$$
I get stuck here, however; I do not know how to proceed. How can I finish this proof?