Given a possibly infinite set $S$ which is closed under the union of two members, ie $x,y\in S\implies x\cup y\in S$, how can I show $S$ is closed under the union of all elements, ie $\bigcup S\in S$?
I can certainly use induction to proof $x_1,\ldots x_n\in S\implies x_1\cup\ldots\cup x_n\in S$ for all $n$ but this doesn't cover the infinite case.
Is it true ?
Take $N$ the set of natural numbers.
Let $X$ be the set of all finite subsets of $N$.
I think this is a counterexample.
(P.S. in general, you can't "prove infinity by induction" you can prove by induction that for all finite $n$ their union is an element, but that's all).