Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$ Let $V$ be a vector space over a field $k$ and let $U,W$ be finite-dimensional subspaces of $V$. Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$.
I know how to prove that $U\cap W$ is a subspace of $V$ but I'm having a hard time grasping how to prove $U+W$
This is what I was doing in regards to the first part of proving closed under addition:
Let $u,w\in U+W$ and $u\in U, w\in W$
Since $U$ is a space, it is closed under addition and $u+w\in U$. Also, $W$ is a space so it is closed under addition as well and so $u+w \in W$. So, $u+w\in W$ and $u+w\in W$ shows that $u+w\in U+W$. Thus, $U+W$ is closed under addition. 
But this is almost exactly how I've proved $U\cap W$ is a subspace and I have a feeling I've made a mistake somewhere. 

There's no reason for $u+w$ to be in $U$ since $w$ doesn't necessarily belong to $U$. 
What you have to prove is that, if $u+w\in U+W$, $u'+w'\in U+W$ ($u,u'\in U$, $w,w'\in W$) then  $(u+w)+(u'+w')$ has the form $(u''+w'')$ for some $u''\in U , w''\in W$, and similarly for $\lambda (u+w)$.
