If a set $S$ has a choice function, does $\bigcup S$ have one too?

set-theory functions axiom-of-choice

No. Of course not. Your proof is perfectly valid.

This is also mentioned in the errata for the book.

Related

Evaluating the limit of a certain definite integral
A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$
Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$
Is a field with cyclic multiplicative group necessarily finite? [duplicate]
What is the volume of the sphere in hyperbolic space?
Literature recommendation on random graphs
Can two random variables $X,Y$ be dependent and such that $E(XY)=E(X)E(Y)$?
Natural equivalence between singular and simplicial homology
another product of log integral
Trilogarithm $\operatorname{Li}_3(z)$ and the imaginary golden ratio $i\,\phi$
find the maximum possible area of $\triangle{ABC}$
How can I show that $X$ and $Y$ are independent and find the distribution of $Y$?