  Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square
$U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and
mutually disjoint rectangles inside $U$ such that the bottom-left corner of
each rectangle coincides with a point in $P_{n}$ and the total area covered by
the rectangles is maximized \cite{ibmpuzzle}, \cite{Winkler2007},
\cite{Winkler2010a}, \cite{Winkler2010b}. The longstanding conjecture has been
that at least half of $U$ can be covered when such rectangles are properly
placed. In this paper, we give an existential proof of the conjecture.
