Math 220
Midterm II
(10 pts) 1 Let g : (0, ∞ ) → (0, ∞ ) be a function defined on the positive real numbers.
Define h : (0, ∞ ) → (0, ∞ ) by h(x) = 1/g(x).
(a) Prove that if g is surjective, then h is surjective.
(b) Prove that if g is injective, then h is injective.
(10 pts) 2 Let x be a real number in the interval [0) 1]. Prove that
(1 - x)2006 ≥ 1 - 2006x.
(Hint: use induction on n to prove the more general inequality where 2003 has been replaced by n.)
(20 pts) 3 Prove that each of the following sets is countable.
(You may use any result proved in class or on the homework; however, you must state clearly exactly what result you are using.)
(a) The set S of real numbers x such that x2 is a natural number; that is S = {x ∈ R|x2 ∈ N}.
(For example, √2 ∈ S and -3 ∈ S, but 2/1 S and π S.)
(b) The set L of all non-vertical lines in R × R such that at least two integer points are on the line; that is L is the set of all lines {y = mx + b} which contain (p1, q1), (p2, q2) ∈ Z × Z with (p1, q1) ≠ (p2, q2).
(For example, the line y = 3x — 5 is in L, since the points (0) —5) and (3) 4) are on it; the line y = πx is not in L, since if x is an integer other than 0, then y = πx is not an integer; the only integer point on this line is (0) 0).)
(10 pts) 4 Find the supremum of the set S = {n/2n−1|n ∈ N} and prove that your answer is correct.