Chapter 3 - CBU

3.2. LIMIT THEOREMS 49Theorem (3.2.7 — Squeeze Theorem). Suppose x n apple y n apple z n 8n 2 Nand lim(x n ) = lim(z n ). Then (y n ) converges andlim(x n ) = lim(y n ) = lim(z n ).Proof. Let w = lim(x n ) = lim(z n ). Given ✏ > 0.9 K 1 2 N 3 8 n K 1 , ✏ < x n w < ✏, and also9 K 2 2 N 3 8 n K 2 , ✏ < z n w < ✏.Let K = max K 1 , K 2 . Then for n K,✏ |{z} < x n w apple y n w apple z n w |{z} < ✏ =) |y n w| < ✏.n K 1 n K 2Thus lim(y n ) = w.Note. The hypotheses of Theorem 3.2.4 thru Theorem 3.2.7 can be weakenedto apply to tails of the sequences rather than to the sequences themselves.Example.⇣ cos n⌘(1) Find lim .nSolution. 1 apple cos n apple 1 =) 1 cos napplen n apple 1 n .⇣1⌘ ⇣ 1Since lim = lim = 0,n n⌘⇣ cos n⌘lim = 0 by the Squeeze Theorem. ⇤n⇤