Theorems on Convergent Sequences with Proofs and Examples

Real Analysis

Lecture-6

Theorems on Convergent Sequence

Dated:-20.05.2020

PPT-18

 UG (B.Sc., Part-2)

Dr. Md. Ataur Rahman

Guest Faculty

Department of Mathematics

M.L. Arya, College, Kasba

PURNEA UNIVERSITY, PURNIA

Theom3:-Every monotonic increasing sequence

                which is bounded above is convergent.

                i.e. converges to its least upper bound.

Proof:-

Let       be a monotonic increasing sequence (m.i.s) and

bounded above.

Since         is m.i.s. hence

Again since           is bounded above, So         has upper bound

i.e. it has some least upper bounds say k

Let the axiom (ii) be hold true for

n=m

     Then

Proof continue

But from (1),

Hence every monotonic increasing sequence which is

bounded above is convergent

Theom4:-Every monotonic decreasing sequence

which is bounded below is convergent.

 i.e. converges to its greatest lower bound.

Proof:-

Let         be a monotonic decreasing sequence (m.d.s)

            and bounded below.

Since           is m.d.s. hence

Again since          is bounded below, So         has greatest lower bound

i.e. it has some greatest lower bounds say M

Let the axiom (ii) be hold true for

n=m

     Then

Proof continue

But from (1),

Hence every monotonic increasing sequence

which is bounded above is convergent.

Examples

  Determine if the the sequence is bounded or not.

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The lecture covers theorems on convergent sequences, including the convergence of monotonic increasing and decreasing sequences when bounded. Detailed proofs for these theorems are provided, along with examples to determine if a sequence is bounded. The presentation includes step-by-step explanations and visual aids.

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Real Analysis Lecture-6 Theorems on Convergent Sequence Dated:-20.05.2020 PPT-18 UG (B.Sc., Part-2) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA

Theom3:-Every monotonic increasing sequence which is bounded above is convergent. i.e. converges to its least upper bound. na Proof:- Let be a monotonic increasing sequence (m.i.s) and bounded above. Since is m.i.s. hence . i e n na 1, a a n N + n n , m a a n m n a Again since is bounded above, So has upper bound i.e. it has some least upper bounds say k n a 0 , Th en for gi ven ( ) ( ii , i a k n N n ) , a k for a t least one va lue of n n Let the axiom (ii) be hold true for n=m Then m a , k

Proof continue But from (1), , , , n m a a So n m a a k n m n m , ....(2) k n N we get n m a k n m n (1), + , From axiom a n N n , a a k k k N n + , .....(3) n n (2) a (3), + from k and ( ) , k m n , ( ).....(1) a k n n = lim n a k n , . So a converges to k n Hence every monotonic increasing sequence which is bounded above is convergent.

Theom4:-Every monotonic decreasing sequence which is bounded below is convergent. i.e. converges to its greatest lower bound. na Proof:- Let be a monotonic decreasing sequence (m.d.s) and bounded below. Since is m.d.s. hence . i e n a 1, a a n n N + n n , ....(1) m a a n m n a Again since is bounded below, So has greatest lower bound i.e. it has some greatest lower bounds say M Then for giv en + n a 0, ( ) ( ) ii a , i a M M n N n , . for a t least one v alue of n n Let the axiom (ii) be hold true for n=m Then + , a M m

Proof continue But from (1), + + , , , n m a a So n n m a a M n m n m , ....(2) m a n N w n m a M n (1), , From the axiom a M a M from M N n N n M n , .... .(3) n n (2) (3 M ) + , and e ge t ( ) , M a m n ( ).....(1) , a M n n = lim n a M n , . So a conv e g r es to k n Hence every monotonic increasing sequence which is bounded above is convergent.