In Cameron's Sets, Logic and Categories (p. 48-49), he sets out prove the following fact about the Zermelo hierarchy $V$, namely, that $V_\alpha\subseteq V_{s(\alpha)}$ for all ordinals $\alpha$. His proof proceeds by induction:
Case 1: $\alpha=0$. Then $V_\alpha$ is the empty set, which is a subset of any set.
Case 2: $\alpha=s(\gamma)$ for some $\gamma$. Take $x\in V_\alpha$. Then, $$x\subseteq V_\gamma\subseteq V_{s(\gamma)}=V_\alpha$$ so $x\in\mathcal{P}V_\alpha=V_{s(\alpha)}$.
Case 3: [limit ordinal case...]
I'm quite confused by Case 2. It seems like Cameron is taking $V_\gamma\subseteq V_{s(\gamma)}$ as the (unstated) inductive assumption. But even then, if $x\in V_\alpha$, how do we know immediately that $x\subseteq V_\gamma$?
By definition, $V_\alpha=V_{s(\gamma)}$ is the power set of $V_\gamma$. So, an element of $V_\alpha$ is the same thing as a subset of $V_\gamma$.