KMB, 22/10/99

M1F Foundations of Analysis—Problem Sheet 3

This is assessed coursework. Please hand in solutions to the starred questions (i.e., to 1a, 1c, 3b, 4, 5 and 6) at the end of the lecture on Monday 1st November.

Reminder: throughout this sheet, if y is a non-negative real number, then \sqrt{y} refers to the non-negative square root of y.

1) Recall the following 5 facts about the ordering < on the real numbers, that were mentioned in lectures: Fact 1: For all reals x and y, exactly one of x < y, x = y and x > y holds. Fact 2: If x, y, z are reals with x < y and y < z, then x < z. Fact 3: If x < y and c ∈ ℝ then x + c < y + c. Fact 4: If x > 0 and y > 0 then xy > 0. Fact 5: If x ∈ ℝ then there is an integer n ∈ ℤ with n ≤ x < n + 1.

Using only these facts about < (but assuming any non-<-related facts about the reals that you need), deduce the following results. State, at each step of your proofs, which fact you are appealing to.

(a)^{∗} If x > 0 and y > 0 then x + y > 0.

(b) If x < y and c < 0 then cx > cy.

(c)^{∗} If 0 < x and x < y then x^{2} < y^{2}.

2) Try proving that if a < b are reals, then there is a rational number c with a < c < b, just using Facts 1–5 from Q1. Hint: Firstly show that there is a large integer N such that N(b − a) > 1.

3) (In this and subsequent questions, you may assume anything reasonable about < and do not have to reduce everything to the Facts stated in Q1.)

(a) Which is bigger, the cube root of 3 or the square root of 2? (This question can be done without using a calculator!)

(b)^{∗} Prove that \sqrt{2} + \sqrt{3} < \sqrt{10}.

4^{∗})

a) Find the set of reals x ≠ 0 such that 2x + \frac{1}{x} < 3.

b) Find the set of reals t such that the equation x^{2} + tx + 3 = 0 has two distinct real solutions x = x_{1} and x = x_{2}.

c) Let x, y, z be reals with x + y + z = 0. Prove that xy + yz + zx ≤ 0.

5^{∗})

a) Consider the real number u = 0.a_{1}a_{2}a_{3}a_{4} …, where a_{i} is the remainder when you divide i by 5. Show that u is rational, and write u as m/n with m, n integers.

b) Consider the real number t = 0.b_{1}b_{2}b_{3}b_{4} …, where

b_{i} = \{ \begin{matrix}1 & if i = 1, 4, 9 … is a square, \\ 0 & otherwise. \end{matrix}

Prove that t is not rational.

6^{∗}) The number 371 is a 3-digit number which is equal to the sum of the cubes of its digits! Prove that there is no 1000-digit number equal to the sum of the 1000th powers of its digits (using a calculator is OK for this one—the trick is working out what to calculate!)