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Mathematics 236, Fall 2022 HW #1 Due September 21
Department of Mathematics and Statistics
PROBLEM 1. Use the principle of mathematical induction (no credit for a solution by any other method) to prove that if x > −1, then (1 + x)n ≥ 1 + nx for all n ∈ N.
PROBLEM 2. A real number that is not rational is said to be irrational.
(i) Proof or counterexample: If x and y are both rational real numbers and x 6= y then x − y is rational.
(ii) Proof or counterexample: If x and y are both irrational real numbers and x 6= y then x − y is irrational.
(iii) Prove that z =√3 −√7 is irrational. (Hint: “eliminate square roots” to find a polynomial with integer coefficients that has z as a root.)
PROBLEM 3. The definition of absolute value of a real number (or more generally, of an element of any ordered field) is given in Section 3 of the text. The symbols a, b, and c denote real numbers.
(a) Show that |b| ≤ a if and only if −a ≤ b ≤ a.
(b) Prove that |a| ≤ |a−b|+|b| and that |b| ≤ |a−b|+|a|. (Hint: (a−b)+b = a and (b−a)+a = b.) Conclude that |a| − |b| ≤ |a − b|, and give an example where the inequality is strict. (c) Use induction to prove that if ak is a real number for each k ∈ N, then |a1 + a2 + · · · + an| ≤ |a1| + |a2| + · · · + |an| ∀n ∈ N
PROBLEM 4. Let A and B be nonempty subsets of R.
(a) Suppose B is bounded above and A ⊂ B. Prove that A is bounded above and sup A ≤ sup B. (b) Suppose A and B are bounded above. Prove sup(A ∪ B) = max{sup A,sup B}.
(c) Define the sum of sets A + B := {x ∈ R : x = a + b for some a ∈ A and b ∈ B}. Suppose A and B are bounded above. Prove sup(A + B) = sup A + sup B. (Hint: it is straightforward to show that the r.h.s. is an upper bound for A + B; you may use the characteristic property of sup to prove it is the smallest one.)
PROBLEM 5. Let S be a nonempty subsets of R.
Suppose f : S → R and g : S → R are bounded functions (this means their ranges are bounded subsets of R.
(a) Show that {f(x) + g(x) : x ∈ S} ⊂ {f(x) : x ∈ S} + {g(x) : x ∈ S}
(b) Prove that sup{f(x) + g(x) : x ∈ S} ≤ sup{f(x) : x ∈ S} + sup{g(x) : x ∈ S}. (c) Give an example of two functions for which the inclusion in part (a) is strict. (d) Show that the inequality in part (b) can be strict.
PROBLEM 6. Compute without proof the suprema and infima (if they exist) of the following sets: (a) {m/n : m, n ∈ N, m < n}; (b) {(−1)m/n : m, n ∈ N, n 6= 0}; (c) {n/(3n + 1) : n ∈ N}; (d) {m/(m + n) : m, n ∈ N \ {0}}.
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