By Hugo D. Junghenn

ISBN-10: 148221928X

ISBN-13: 9781482219289

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Such that c0 ≤ ck−1 ≤ ck ≤ ank ≤ dk ≤ dk−1 ≤ d0 , and dk+1 − ck+1 = 21 (dk − ck ). Since {ck } and {dk } are monotone and bounded ck → c and dk → d for some c, d ∈ R. Since dk − ck = 2−k (d0 − c0 ) → 0, c = d. By the squeeze principle, ank → c. I0 c0 I1 c 1 I2 I3 d0 an1 an2 c2 c3 d2 an3 .. 3: Interval halving process. 5 Theorem. Every sequence in R has a subsequence that converges in R. Proof. If {an } is bounded, then the Bolzano–Weierstrass theorem applies. Suppose that {an } is unbounded above.

The set {n/(n + 1) : n ∈ N} is bounded above by 1 and below by 1/2. 2 Definition. Let A be a nonempty subset of an ordered field F. An upper bound u0 of A with the property that u0 ≤ u for all upper bounds u of A is called a least upper bound, or supremum, of A, and is denoted by sup A. A lower bound 0 of A such that ≤ 0 for all lower bounds of A is called a greatest lower bound, or infimum, of A, and is denoted by inf A. If sup A ∈ A, then sup A is called the maximum of A. If inf A ∈ A, then inf A is called the minimum of A.

F) If ab = 0, then either a = 0 or b = 0. (g) If ab = ac and a = 0, then b = c. (h) If b = 0 and d = 0, then a/b = c/d if and only if ad = bc. (i) If a = 0 and b = 0, then (ab)−1 = a−1 b−1 , or 1 1 1 = . ab a b Proof. (a) If a + 0 = a for all a then, in particular, 0 + 0 = 0. But, by definition of 0 and commutativity of addition, 0 + 0 = 0 . Therefore 0 = 0. (b) By associativity and commutativity of addition, b = b + 0 = 0 + b = (−a + a) + b = −a + (a + b) = −a + 0 = −a. (c) If a · 1 = a for all a then, in particular, 1 · 1 = 1.

### A course in real analysis by Hugo D. Junghenn

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