Chapter 3 - CBU

3.5. THE CAUCHY CRITERION 67⇣Problem (Page 91 # 2b). Show 1 + 1 2! + 1 3! + · · · + 1 ⌘is Cauchy.n!Proof.hIn this proof we use the facts that 2 n 1 apple n! (Example 1.2.4(e)) and that1 + r + r 2 + · · · + r n = 1 irn+11 r . Given ✏ > 0. WLOG, suppose n m.|x n x m | =⇣1 + 1 2! + 1 3! + · · · + 1 n!1(m + 1)! + 1(m + 2)! + · · · + 1 n!1⇣1 + 1 2 m 2 + · · · 1⌘2 n m 1⌘⇣1 + 1 2! + 1 3! + · · · + 1 ⌘m!apple 12 m + 12 m+1 + · · · + 12 n 1 apple= 12 · 1 12mn112m= 2n m 12 n 1 apple2 n m2 = 1n 1 2 < ✏ (= 1 m 1 ✏ < 2m 1 (= ln ✏ln 2 < m 1 (= 1 ln ✏ln 2 < m.n hln ✏i oChoose H = max 1, 1 + 1 . Thenln 2n m H =) m > 1ln ✏ln 2 =) |x n x m | < ✏.Definition (3.5.7). A sequence X = (x n ) is contractive if9 0 < C < 1 3 |x n+2 x n+1 | apple C|x n+1 x n | 8 n 2 N.C is the constant of the contractive sequence.=⇤