Bounded Sequences in Real Analysis

Real Analysis Lecture-4 Sequence Dated:-16.05.2020 PPT-13 UG (B.Sc., Part-2) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA

Bounded Sequence Bounded above :-Any sequence is said to be bounded above if there exists a number k such that Where the number k is called the upper bound of the sequence Example 1, , 2 3 4 n na = , . . . 1,2,3,.... na k n N ie n n a 1 1 1 1 1 n = = , , , , 0 ( ) Let a when n n 1 n 1 1 , 4 3 1 2 0 , ,, , 1 lower b ound upper bound 1, H ere a n N n So, is bounded above and 1 is called upper bound of the sequence n a . n a

Bounded Sequence Bounded below :-Any sequence is said to be bounded below if there exists a number h such that Where the number h is called the lower bound of the sequence Example 1, , 2 3 4 n na = , . . . 1,2,3,.... na h n N ie n a n 1 1 1 1 1 n = = , , , , 0 ( ) Let a when n n 1 n 1 1 , 4 3 1 2 0 , ,, , 1 lower b ound upper bound 0, H ere a n N n So, is bounded below and 0 is called lower bound of the sequence n a . n a

Bounded Sequence Definition :-Any sequence is said to be bounded if there exists two numbers h and k such that Where h is called the lower bound and k is called upper bound of the sequence Example , n Let a then n na = , . . . 1,2,3,.... h a k n N ie n n a n = 1 1 1 1 , , 2 3 4 1 n = 1, , , , 0 a n 1 n 1 1 , ,, 4 3 1 2 0 , , 1 lower bound upp er boun d 0 1, Here a n N n is a bounded sequence, n a

Least upper bound (L.u.p) Definition:-Any sequence is said to be have least upper bound if there exists a number k such that then the number k is called the least upper bound of the sequence Example 1, , 2 3 4 1 n Here Also a But is the smallest uppe na ( ) ( ) ii a , i a k n N n , k for at least one value of n n n a 1 n 1 1 1 1 n = = , , , , 0 ( ) Let a when n n 1 1 , 4 3 1 2 0 , ,, , 1 lower bound upper bound 1 1.1,1.2,...,2,2.1,2.2,... , a n N n all are upper bounds of a r bound of n n a 1 n So, here 1 is called least upper bound of the sequence . n a

Greatest Lower bound (G.l.b) Definition:-Any sequence is said to be have greatest lower bound if there exists a number h such that then the number h is called the greatest lower bound of the sequence Example 2 3 4 1 0 n a n N na + ( ) ( ) ii a , i a h n N n , h for at least one value of n n n a 1 n 1 1 1 , 1 n = = 1, , , , , 0 Let a n 1 1 , 4 3 1 2 , ,, , 1 lower bound upper bo und 0, n 1.1, 1.2,..., 2, 2.1, 2.2,... is the greatest lowe Also a But all are low er bounds of a n n a 0 r bound of n So, here o is called greatest lower bound of the sequence . n a

BOUDED SEQUENCES DEFINITION } { an M DEFINITION } { an M If M is an upper bound but no number less than M is an upper bound then M is the least upper bound. If m is a lower bound but no number greater than m is a lower bound then m is the greatest lower bound n a n a bounded from above bounded from below for all for all M n M n Lower bound Upper bound +n 1 Is bounded below Example greatest upper bound = ?? 3 Example +1 n 3 n a n Is bounded above by any number greater than one If is not bounded we say that n a n a If is bounded from above and below, 1 . 1 = . 1 001 n a M n a n a n a 1 unbounded bounded Least upper bound

Examples Determine if the the sequence is bounded or not. + + + + 3 1 1 2 3! 1! n n n n ( ) i ( ) ii