Proof that dim(ran(T + S)) ≤ dim(ran(T)) + dim(ran(S))

Question

Suppose V and W are vector spaces and assume W is finite-dimensional. Let T and S be linear transformations mapping V into W. Prove: dim(ran(T + S)) ≤ dim(ran(T)) + dim(ran(S)).

I simplified it down to dim((T+S)[V])≤ dim(T[V])+dim(S[V]). Then if I prove that (T+S)[V] is a subset of T[V]+S[V] and vice versa it should complete the proof. However, this is where I get stuck. I have: Let x in (T+S)[V] NTS x in T[V]+S[V]. Where do I go from here?

Answer 1

If $x\in(T+S)(V)$, then $x=(T+S)(v)$, for some $v\in V$. Therefore$$x=T(v)+S(v)\in\operatorname{range}(T)+\operatorname{range}(S).$$