This is exercise 7.1.6 of the book Introduction to Set Theory 3rd ed. by Hrbacek and Jech.
Let $h^{*}(A)$ be the least ordinal $\alpha$ such that there exists no function with domain $A$ and range $\alpha$. Prove:
(a) If $\alpha \geq h^{*}(A) $, then there is no function with domain $A$ and range $\alpha$.
(b) $h^{*}(A)$ is an initial ordinal.
(c) $h(A)\leq h^*(A)$ ($h(A)$ denotes the Hartogs's number of $A$)
(d) If $A$ is well-orderable, then $h(A)=h^*(A)$.
(e) $h^*(A)$ exists for all $A$.
Questions:
I guess the authors meant to define $h^{*}(A)$ as the least nonzero ordinal $\alpha$ such that there exists no function with domain $A$ and range $\alpha$?
I don't see how to show (a) without using a constructive definition of $h^{*}(A)$ (as in Hartogs's theorem):
Let $\beta=\{\alpha:\alpha \text{ is an ordinal isomorphic to some well-ordered partition of } A\}$. The set $\beta$ exists by the replacement axiom. Moreover, $\beta$ is precisely the set of ordinals $\alpha$ such that there exists a function with domain $A$ and range $\alpha$. Then $h^{*}(A):=\beta\cup\{0\}$ is the least nonzero ordinal $\alpha$ such that there exists no function with domain $A$ and range $\alpha$.
It seems we have to answer (e) before (a)?
For 1, yes that's correct, they should have stipulated $\alpha\ne 0.$
For 2, we can just redefine a surjection $g: A\to \alpha$ as $h(a)= g(a)$ if $g(a)<h^*(A)$ and $h(a)=0$ otherwise, and $h$ is a surjection $A\to h^*(A).$
So we only need (e) for (a) in the sense that they should have prefaced (a) with "if $h^*(A)$ exists" to make it make sense before we have proven (e).