I have to prove the following for a vectorspace $V$ with $dim(V)=n<\infty$ and several subspaces $U_i$ $$V=U_1\oplus U_2 \oplus ..... \oplus U_k \iff dim(U_1) + dim(U_2) + ... + dim(U_k) = n$$
I am not quite sure how to do this. I have looked into the theorems I learned, but those did not help me very much. Any hints?
Seems to be true if $k=1$. Suppose it's true for all $k$ up to $n-1$. Rank-nullity gives that it's true for $k=n$ by projecting onto the $n^\text{th}$ subspace.