Question: Suppose $U_1, \cdots, U_m$ are finite dimensional subspaces of $V$. Prove that $U_1 + \cdots + U_m$ is finite dimensional, and that $\dim(U_1 + \cdots + U_m) \leq \dim U_1 + \cdots + \dim U_m$.
My solution: I attempted an induction proof with the identity $\dim(A + B) = \dim A + \dim B - \dim (A \cap B)$.
Clearly $\dim A \leq \dim A$. Then hypothesize that $\dim(U_1 + \cdots + U_m) \leq \dim U_1 + \cdots + \dim U_m$.
Add $U_{m+1}$ to both sides and use the identity and inductive hypothesis:
\begin{align} \dim(U_1 + \cdots + U_m + U_{m+1}) &= \dim(U_1 + \cdots + U_m) + \dim U_{m+1} \\&\hspace{2cm}- \dim((U_1 + \cdots+ U_m) \cap U_{m+1}) \\ &\leq \dim(U_1 + \cdots+ U_m) + \dim U_{m+1} \\&\leq \dim U_1 + \cdots + \dim U_{m+1} \end{align}
I'm just looking for a solution verification because the solution manual lists a completely different answer.