In Paul Halmos "naive set theory" he has a long proof for the recursion theorem. The recursion theorem is
If $a$ is an element of a set $X$, and if $f$ is a function from $X$ into $X$, then there exists a function $u$ from $\mathbb{N}$ into $X$ such that $u(0)=a$ and sucht that $u(n^+)=f(u(n))$ for all $n$ in $\mathbb{N}$.
I would just prove it by defining $u$ to be $f^{(n)}(a)$, such that $f^{(0)}(a)=a, f^{(1)}(a)=f(a), f^{(2)}(a)=f(f(a))$, etc.
But in the book the proof is much longer and more complicated. Is what I wrote a legal proof?, if not, why is it not correct?