I've been independently reading Kunen's newest set theory book for a self-study course. I'm looking at his chapter on cardinal arithmetic, thoroughly reading proofs and working on exercises. After an exercise of the Milner-Rado Paradox, he mentions that types are bounded when it comes to finite unions and states the following exercise:
Let $\kappa$ be an infinite cardinal and $\alpha = \bigcup_{n<c}X_n$, where $c < w$ and type$(X_n) < \kappa^{\omega}$, then $\alpha < \kappa^{\omega}$.
Intuitively, it seems to make sense, but I'm having trouble coming up with a formal proof. Would I use induction on $n$? Any help or hints would be much appreciated. Thanks!
You have to prove that if $\operatorname{type}(X_i)<\kappa^\omega$ for $i=1,2$, then $\operatorname{type}(X_1\cup X_2)<\kappa^\omega.$
Visualize $X_1$ as an ordinal $\alpha<\kappa^\omega$. When you add $X_2$ to $X_1$, for each $\beta<\alpha$ you're adding between $\beta$ and $\beta+1$ a set of type at most $\operatorname{type}(X_2)$, and as you're doing this $\operatorname{type}(X_1)$ times, you get that $\operatorname{type}(X_1\cup X_2)\leq\operatorname{type}(X_2)\cdot\operatorname{type}(X_1)<\kappa^\omega.$