Let R be the rectangle shown here, including the edges. Let H be the set of all rectangles whose sides have positive length, are parallel to the sides of R, and lie within R. H is partially ordered by set inclusion.
(a) Does every subset of H have an upper bound? a least upper bound?
(b) Does every subset of H have a largest element?
(c) Does every subset of H have a lower bound?
(d) Does every subset of H have a smallest element?
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