Let $F$ be family of function for which $f,g \in F \rightarrow f \subseteq g$ or $g \subseteq f$. I need to prove that $\cup F$ is a function.
The result is very obvious, but how the proof should go is not. For example, is it allowed to pick the function with the greatest cardinality in the family and then show that $\cup F$ is that function? What kind of proof technique should be used?