Originally Posted by

**vincisonfire** Hi,

Suppose that V is a vector space and $\displaystyle \{W_i : i \in I\} $ is a nonempty collection of subspaces of V.

I want to show that if $\displaystyle \{W_i : i \in I\} $ is a chain, then $\displaystyle \cup_{i \in I} W_i $ is a subspace of V.

Intuitively, this is obvious but I don't know how to prove that $\displaystyle \{W_i : i \in I\} $ is closed under union of chains.

Then I think I could use Zorn's lemma to say that it has a maximal element that is a subspace of V.