Here is Prob. 8, Sec. 10, in the book Topology by James R. Munkres, 2nd edition:
Problem 8 (a):
Let $A_1$ and $A_2$ be disjoint sets, well-ordered by $<_1$ and $<_2$, respectively. Define an order relation on $A_1 \cup A_2$ by letting $a < b$ either if $a, b \in A_1$ and $a <_1 b$, or if $a, b \in A_2$ and $a <_2 b$, or if $a \in A_1$ and $b \in A_2$. Show that this is a well-ordering.
Problem 8 (b):
Generalize (a) to an arbitrary family of disjoint well-ordered sets, indexed by a well-ordered set.
I think I'm clear as to the proof required in Prob. 8 (a).
My Attempt at Prob. 8 (b):
Let $J$ be a (non-empty) well ordered set; let $$\left\{ \ A_\alpha \ \colon \ \alpha \in J \ \right\}$$ be a collection of non-empty, (pairwise) disjoint well-ordered sets indexed by set $J$; and let $$ A \colon= \bigcup_{\alpha \in J} A_\alpha. $$ For each $\alpha \in J$, let $<_\alpha$ denote the well-ordering relation on the set $A_\alpha$.
For any two elements $a, b \in A$, let us define $a < b$ to mean the following:
Either $a, b \in A_\alpha$ for some $\alpha \in J$ and $a <_\alpha b$, or $a \in A_\alpha$, $b \in A_\beta$ for some $\alpha, \beta \in J$ such that $\alpha <_J \beta$.
Then the set $A$ is a well-ordered set.
Is this statement correct?
If so, then here is my proof:
Let $S$ be a non-empty subset of the set $A = \bigcup_{\alpha \in J} A_\alpha$. Let $J_0$ be the following subset of $J$. $$ J_0 \colon= \left\{ \ \alpha \in J \ \colon \ S \cap A_\alpha \neq \emptyset \ \right\}. $$ Then the set $J_0$ is non-empty, and as such it has a smallest element, say $\alpha_0$. Then $\alpha_0 \in J$ and $S \cap A_{\alpha_0}$ is a non-empty subset of $A_{\alpha_0}$ and so has a smallest element $a_{\alpha_0}$, which is also the smallest of $S$ with respect to the order relation on $A$.
Is this proof correct? If so, then is my presentation accessible enough? If not, then where have I erred?