Assignment 1 using our definition of multiplication in

Question

Assignment 1-

1. Prove that the set of rational numbers

{x ∈ Q : x³ < 2}

is a real number. (This is what we would identify as 3√2.)

2. Using our definition of multiplication in class, show that if x and y are real numbers, then so is x · y.

3. Let s be an arbitrary real number.

(a) Based on s, what should the definition of -s be? Give an appropriate definition of -s, and prove that your definition of -s indeed makes it a real number.

(b) Prove that for any real number s we have that s + (-s) = 0.

4. Find, with full justification, the sup and inf of each of the following sets, or prove they don't exist.

(a) {2^1/2, 2^1/4, 2^1/8, . . .}

(b) {1 + (-2)ⁿ: n ∈ Z}

(c) {1 + ((-1)ⁿ/3n): n ∈ Z}

5. Suppose A and B are sets of real numbers. Prove that if A is a subset of B, then sup B ≥ sup A ≥ inf A ≥ inf B.