Chapter 3 - CBU

3.1. SEQUENCES AND THEIR LIMITS 37Theorem (3.1.5). Let X = (x n ) be a sequence in R, and let x 2 R. Thefollowing are equivalent:(a) X converges to x.(b) 8 ✏ > 0, 9 K 2 N 3 8 n K, |x n x| < ✏.(c) 8 ✏ > 0, 9 K 2 N 3 8 n K, x ✏ < x n < x + ✏.(d) 8 ✏-nbhd. V ✏ (x) of x, 9 K 2 N 3 8 nProof.(a) () (b) by definition.(b) () (c) () (d) sinceK, x n 2 V ✏ (x).|x n x| < ✏ () ✏ < x n x < ✏ () x ✏ < x n < x + ✏ ()x n 2 V ✏ (x).TechniqueGiven ✏ > 0. Produce or verify the existence of an integer K(✏) so thatn K(✏) =) |x n x| < ✏.Sometimes |x n x| < ✏ can be converted, with reversible steps, to an inequalityof the form n > f(✏). Take K(✏) as the first integer greater than f(✏) (by theArchimedean Property), K(✏) = [f(✏)] + 1, for example. ThenExample.n K(✏) =) n > f(✏) =) |x n x| < ✏.(1) lim(c) = c, c 2 R, i,e., x n = c 8n 2 N.Proof. Given ✏ > 0. [To show 9 K(✏) 2 N 3 8 n K(✏), |c c| < ✏.]|c c| = 0 < ✏ 8 n 2 N. Pick K(✏) = 1.Then n K(✏) =) |c c| < ✏. ⇤⇤