I identify the natural number $0$ with the empty set $\emptyset$, $1$ with $S(0)$, $2$ with $S(1)$, etc, etc.
The axiom of infinity says $\exists x (\emptyset\in x\wedge \forall z\in x\space z\cup\{z\}\in x)$ and the Axiom schema of specification says $\forall y_0,...,y_n\exists x\forall z (z\in x\leftrightarrow (z\in y_0\wedge \phi(z,y_1,...,y_n)))$.
My question now: Why is there now a smallest element $x$ which can be identified with the natural numbers?
Let $y$ be an inductive set whose existence follows from the axiom of infinity, then consider $\{x\subseteq y\mid x\text{ is inductive}\}$. This is a definable collection of members of the power set of $y$, so it is a set, and $y$ is there so it's not an empty set.
Now take the intersection of all those sets. This is an inductive set as well (you have to prove this, of course). Call this inductive set $N$. Now prove that if $x$ is any inductive set then $N\subseteq x$, by considering $M=N\cap x$. Show that $M$ is inductive as well, and $M\subseteq y$, now use the property which defined $N$ to conclude $N=M$ and therefore $N\subseteq x$.
And now we're done.