Let $E$ be a vector space and $F$ a subspace of $E$. Show that if $E$ is finite dimensional then $F$ is finite dimensional too.
It's easy to prove by contradiction by taking a family linearly independent with $n+1$ vectors of $F$.
I can also prove it by using axiom of choice, I am curious if we can avoid axiom of choice to prove the result 'directly' (by directly I mean without using the method by contradiction)? Thanks