Let $W_1,\ldots,W_n$ be subspaces of a vector space $V$. Prove that there exist subspaces $U_1 \subseteq W_1, \ldots , U_n \subseteq W_n$ such that $W_1 + \cdots + W_n = U_1 \oplus \cdots \oplus U_n$. (direct sum)
Should I start by defining a basis for each $W_i$? I could see where I could obtain a basis for $W_1 + \cdots + W_n$ by taking the union of the bases for each $W_i$ and then eliminating dependent elements.
I know that for $W_1 + \cdots + W_n$ to be the direct sum of the $U_i$'s, I need the intersection of each $U_i$ with the rest to be zero.
Just not sure how to rigorously start this proof or how to go about defining the $U_i$'s.