Prelims Analysis Oxford, Exercises of Mathematics

Download Prelims Analysis Oxford and more Exercises Mathematics in PDF only on Docsity! Analysis I Sheet 1 — MT21 Real Numbers, Arithmetic, Order, (Real) Modulus 1. Prove, from the given axioms for the real numbers, that for real numbers a, b, c, d (a) a(bc) = c(ba); (b) −(a+ b) = (−a) + (−b); (c) if ab = ac and a ̸= 0, then b = c; (d) if a < b and c < d then a+ c < b+ d; (e) if a ⩽ b and c ⩽ d then a+ c = b+ d only if a = b and c = d. [Try to write out detailed answers, justifying each line of your argument by appealing to one of the axioms.] 2. Prove the following assertions, for real numbers a, b, c: (a) if a < b, then ac > bc if and only if c < 0; (b) a2 + b2 = 0 if and only if a = b = 0; (c) a3 < b3 if and only if a < b. [Less detailed answers are required than in Q1, but you should try to justify each step using axioms or results which have already been proved from the axioms.] 3. (a) Prove that (am)−1 = (a−1)m, for all a ∈ R \ {0} (for m = 1, 2, . . . ). (b) Prove that ak+1 = aka for a ̸= 0 and k = −1, −2, −3, . . . . (c) Derive the law of indices: aman = am+n for a ̸= 0 and m, n ∈ Z. Mathematical Institute, University of Oxford Page 1 of 2