This is a fairly popular analysis book with many, many questions and no answers. So I will gradually attempt to answer some of them here. I may be wrong sometimes, so feel free to correct me if you can. If you are using this to cheat, then shame on you (this blog entry became the second most viewed after just one week; what's up with that?)!
Claim:
Let be rational. Then , where , , and . This last assumption can be made without loss of generality. It follows that
Nevertheless, this last expression is absurd; an odd number cannot be equal to an even number. Therefore, is irrational. This proves the claim.
Claim: If for every
Let for every . Then the hypothesis holds. Moreover, let . However, . Therefore, . This dismisses the claim.
Claim: If for every
Let, for every , , be a nonempty set containing a finite number of real numbers, and . Then there exists an such that for every . Therefore, , which is a nonempty set containing a finite number of real numbers. This proves the claim.
Claim:
Let , , and . Then , and , which are different. Therefore, the claim is dismissed.
Claim:
Let . Then or . implies that or . or implies that . or implies that . Therefore, . The converse can be proved in the same manner.
Claim: If
Lemma: , , .
Given a function and a subset , let be the set of all points from the domain that get mapped into ; that is, . This set is called the preimage of .
Let . If is the closed interval and is the closed interval , find and .
Does in this case?
Therefore, they are equivalent.
The good behavior of preimages just demonstrated is completely general. Show that for an arbitrary function , it is always true that and for all sets .
Let . Then . This implies that and . This implies that and . This implies that .
To be continued...