Chapter 3 - CBU

3.4. SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM 61Theorem (3.4.4). Let X = (x n ) be a sequence. The following are equivalent:(a) (x n ) does not converge to x 2 R.(b) 9 ✏ 0 > 0 3 8 k 2 N, 9 n k 2 N 3 n k k and |x nk x| ✏ 0 .(c) 9 ✏ 0 > 0 and a subsequence X 0 = (x nk ) of X 3 |x nk x| ✏ 0 8 k 2 N.Proof.[(a) =) (b)] This is the negative of the definition of convergence.[(b) =) (c)] Take the ✏ 0 from (b).Let n 1 2 N 3 |x n1 x| ✏ 0 .Let n 2 2 N 3 n 2 > n 1 and |x n2 x| ✏ 0 .Let n 3 2 N 3 n 3 > n 2 and |x n3 x| ✏ 0 .Continuing, we generate the subsequence.[(c) =) (a)] Suppose X = (x n ) has a subsequence X 0 = (x nk ) satisfying (c).If x n ! x, so would (x nk ) ! x. Then 9 K 2 N 3 8k K, |x nk x| < ✏ 0 .But this contradicts (c).⇤