Chapter 3 - CBU

3.4. SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM 63Theorem (3.4.9). Let X = (x n ) be a bounded sequence such that everyconvergent subsequence converges to x. Then lim(x n ) = x.Proof. Let M be a bound for X. Suppose x n 6! x. By Theorem 3.4.4,9 ✏ 0 > 0 and a subsequence X 0 = (x nk ) 3 |x nk x| ✏ 0 8 k 2 N.Now M is also a bound for X 0 = (x nk ),so it has a convergent subsequence X 00 = (x nkr ) with lim(x n k r ) = x.Then 9 K 2 N 3 8 r K, |x nkrx| < ✏ 0 , a contradiction. ⇤Example. We cannot drop the bounded hypothesis:⇣1, 1 2 , 3, 1 4 , 5, 1 6 , . . . ⌘.