Let R be the rectangle shown here, including the edges. Let H be the set of all rectangl...

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?