Need help on problems for Elementary Analysis class. (Proof based Calc class)?

Question

There are several problems, if you can help out on some (feel free to solve but even just tips on where to start or how to go about them) I'd greatly appreciate it. In answers, please put the number of which you're responding to. Thanks in advance!

2. If A, not equal to empty set, is bounded below, let B be the set of all lower bounds of A. Show that B is not equal to empty set, that B is bounded above, and that sup B is the greatest lower bound of A.

8. For which numbers a, b, c, and d will the function: F(x) = (ax +b)/(cx+d) satisfy f(f(x)) = x for all x such that f(f(x)) is defined.

13.
a. Prove that any function f with a domain of all reals can be written as f = E + O, where E is even and O is odd.
b. Prove that this way of writing f is unique.

Answer 1

2. I'm assuming we're working with sets of real numbers. Since A is bounded below, there is a real number b so that b <= a for all a in A. That tells us that B cannot be empty; the fact that A is bounded below tells us there exists a lower bound. Suppose B is not bounded above. Then, for some b in B, there is no number a such that b <= a. However, b <= a for all a in A, so that is a contradiction. Finally, by the completeness property of the real numbers, B has a supremum, and A has an infinum. Try showing that they must be equal, or else you can arrive at a contradiction.

8. f(f(x)) = [a(ax+b)/(cx+d) + b] / [c(ax+b)/(cx+d) + d]. Simplify by multiplying both the numerator and the denominator by cx+d. Then see how to choose a,b,c,d so that the result is x.

13.a. E=[f(x)+f(-x)]/2. O can be defined similarly. You can verify that E and O are odd and even using the definitions of these terms.