I have a proof for showing that a subspace of finite dimensional vector space if finite dimensional and I'm not sure if it works. If $W$ is a subspace of the vector space $V$, dim(V)=n and we assume dim(W) is infinite. $V$ has to contain the spanning set of $W$ for $W$ to be a subspace of $V$, however if dim(W) is infinite it doesn't have a finite spanning set. This means $V$ doesn't have a finite spanning set either, contradicting that dim(V)=n because n is finite. Does this proof work?