1. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom Example: (1)lub (−∞, 0) = 0, lub(−∞, 0] = 0 (2) lub (−4,−1) = −1, lub(−4,−1] = -1 (3) lub {1/2, 2/3, 3/4,..., n/(n + 1),... } = (4) lub {−1/2,−1/8,−1/27,...,−1/n 3,... } =

2. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. (5) lub {x : x 2 < 3} = lub{x : } (6) For each decimal fraction b = 0.b 1 b 2 b 3,..., we have b = lub {0.b 1, 0.b 1 b 2, 0.b 1 b 2 b 3,... }. (7) If S consists of the lengths of all polygonal paths inscribed in a semicircle of radius 1, then lub S = π (half the circumference of the unit circle).

3. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom

4. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom Example (a)Let S = {1/2, 2/3, 3/4,..., n/(n + 1),... } and take ε = 0.0001. (b) Let S = {0, 1, 2, 3} and take ε = 0.00001.

5. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom

6. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Least Upper Bound Axiom

7. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers

8. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers For example:

9. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers From a 1, a 2, a 3,..., a n,... and b 1, b 2, b 3,..., b n,... we can form the scalar product sequence : αa 1, αa 2, αa 3,..., αa n,..., the sum sequence : a 1 + b 1, a 2 + b 2, a 3 + b 3,..., a n + b n,..., the difference sequence : a 1 − b 1, a 2 − b 2, a 3 − b 3,..., a n − b n,..., the product sequence : a 1 b 1, a 2 b 2, a 3 b 3,..., a n b n,.... If b i ≠ 0 for all i, we can form the quotient sequence : the reciprocal sequence :

10. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers The sequence with terms a n is said to be increasing if a n < a n+1 for all n, nondecreasing if a n ≤ a n+1 for all n, decreasing if a n > a n+1 for all n, nonincreasing if a n ≥ a n+1 for all n. A sequence that satisfies any of these conditions is called monotonic.

11. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The sequences 1, ½, 1 ∕3, ¼,..., 1 ∕n,... 2, 4, 8, 16,..., 2 n,... 2, 2, 4, 4, 6, 6,..., 2n, 2n,... are monotonic. The sequence 1, ½, 1, 1 ∕3, 1, ¼, 1,... is not monotonic.

12. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sequences of Real Numbers Example The sequence is increasing. It is bounded below by ½ (the greatest lower bound) and above by 1 (the least upper bound).

13. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.