Decide whether the following statements are true or false. Provide counter examples for those that are false, and supply proofs for those that are true. a. An open set that contains every rational number must necessarily be all of R. b. The Nested Interval Property remains true if the term "closed interval" is replaced by "closed set." c. Every nonempty open set contains a rational number. d. Every bounded infinite closed set contains a rational number. e. The Cantor set is closed.