For each of the following universal assertions about subsets of R, prove it (if true) or give a counterexample (if false) and explain why it is a counterexample. a) Every bounded nonempty open subset of R is an open interval. b) Every closed nonempty subset of R contains its maximum. c) Every bounded closed nonempty subset of R contains its maximum. d) Every nonempty open subset of R is a union of open intervals.