1 Br. Bradley Quoc Phung 230 A Sequences Definition: A sequence sn converges to L 0, n N n N d sn , L n converges to 1 n n Example: Prove Sn = Solution: Choose 0 , n 1 n n Sn 1 n n 1 n n n n n n n n n 1 n 1 1 n n 2 Then If we choose n For N 1 2 1 2 , then n n 1 1 1 n n n n n , n N Sn 1 . Therefore : Sn = n converges to 1 n n 2 Br. Bradley Quoc Phung 230 A 1. Theorem: Convergent sequences are bounded Proof: Suppose Sn converges to L. Choose = 1, then N ie: lim Sn L such that n > N Let r = max { 1, d ( S1 , L ) , d ( S2 , L ) x d ( Sn , L ) < = 1 …. , d ( Sn , L ) } Then take the ball B( L, r+1). This will contain all terms of the sequence. Sequence Sn is bounded. Note: 1. Intervals of the form (a , a ) {x R : x a .} i.e: The interval an - neighborhood of a. Denote (a -r, a+ r ) by the open ball B(a, r) centered at a with radius r B(a, r) = x R n : x a r 2. A subset U of Rn is open set a U, there is some r = r(a)> 0 so that U B(a, r) A set S R n is bounded r > 0 such that S B( o, r) 3 Br. Bradley Quoc Phung 230 A 2. Theorem: If Sn converges to s Sn converges to t Then s = t Proof: Since Sn 2 converges to t, then N1 such that n > N2 | Sn - t | < 2 Since Sn converges to s, then N1 such that n > N1 | Sn - s | < Let N = max{ N1, N 2 } Then n > N | s-t | = | s- Sn + Sn – t | | s- Sn | + | Sn – t | = + 2 2 = This is true for any choice of s = t. Theorem: A real sequence Sn converges to s, then every open ball B ( s , r ) contains all , but a finite members of terms form the sequence. Definition Non-decreasing Of Sequence A real sequence Sn is non- decreasing n N, S n + 1 S n. A real sequence Sn is non- increasing n N, S n + 1 S n. A real sequence Sn is monotone non- decreasing or non- increasing. 4 Br. Bradley Quoc Phung 230 A Theorem: A monotone real value sequence is convergent It is bounded Proof: Convergent Bounded. Bounded Convergent. Suppose Sn is a bounded monotone. Let Sn be Non-decreasing. Since Sn is bounded, the set S = { S n | n S has a suppremum. Let L = sup S. From definition of suppremum, Given > 0 , element sn* S such that Since Sn is Non-decreasing, 0, n* such that n > n* ie: Sn = 1,2,3,4,.. }is bounded above. L sn* L if n > n* , then sn sn* L sn L L L sn L d ( L, s n ) < converges to L. Note: All we need have is that the sequence is eventually bounded and monotone.