Let's take $K$ to be an infinite field and $V$ a vector space over it. Allowing $U_1,...U_l$ to be $l$ proper subspaces of $V$ (none the same), and further assuming that for all $i$, $U_i$ is not in the union of the $U_j$ for $i\neq j$, I want to show that there exists a $v\in V$ not in any of the $U_i$.
Thus far, my course of action has been induction on $l$. The base case is trivial, and then for the induction step I let $v_1\in V$ be such that it's not in $U_i$ for $2\leq i\leq l+1$ and $v_{l+1}\in V$ such that it's not in $U_i$ for $1\leq i \leq l$. I want now that $v=v_1+av_{l+1}$ for $a\in K$ is the vector not in any of the $U_i$, but I'm stuck trying to show this. I'm assuming the hypothesis that $K$ is infinite comes into play but I can't quite see how.
Additionally, what more precise statement can I make in the case that $K$ is finite? Surely if $|K|\gt l$ then I can be more precise about the $a$ above?