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By Manfred Stoll

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But if a 6 B, then since B contains no largest 24 Chapter 1 The Real Number System element, there exists q E B such that q > a. This contradicts the fact that a is an upper bound of B. Similarly, if a2 > 2, then there exists a q < a such that q2 > 2. 3. 5, Exercise 9), which we know is not rational. Least Upper Bound Property of R The following property, also referred to as the completeness property of R. distinguishes the real numbers from the rational numbers and forms the foundation for many of the results in real analysis.

Prove the following. a. If a > b, then a + c > b + c. c. If a * 0, then a2 > 0. 4. *If a, b E IR, prove that ab b. If a > b and c < 0, then ac < bc. d. If a > 0 then I/a > 0, and if a < 0 then 1/a < 0. ](a2 + b2). 4 The Least Upper Bound Property 27 5. Find the supremum and infimum of each of the following sets. }= 248 1l 'c. E{2nn:nEt`11 "g. Bsinn 22 :nEN} ll J d. D {cos ni :nEN} f. :nGN} 1( h. l={xER:xz<4} 6. If 0 < y < 1, prove that there exists a unique positive real number x such that x2 = y.

This, however, is a contradiction. , the sequence diverges. (e) As our final example we prove that li(Vn+ I - Vn) = 0. We first note that ( n+1-y) (Vn + I +V) 1 ( n+1+\) I n+1+Vn1 2Vn- Given e > 0, we want to choose no E NI such that 1/(2\) < e for all n y n,. This is easily verified to be the case if n, E N is chosen so that n, a 1/4E2. With this choice of n, we now have I n + 1 - VnI < e for all it at n,. 9 DEFINITION A sequence { in R is said to be bounded if there exists a positive constant M such that M for all it E N.

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