Reference no: EM131084793
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) {21/2, 21/4, 21/8, . . .}
(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.
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