Explain the key points of 11.3.6 THEOREM 11.3.6A nondecreasing sequence which is bounded above converges to the leastupper bound of its range.A nonincreasing sequence which is bounded below converges to the greatestlower bound of its range.

Theorem 11.3.6: Convergence of Monotonic SequencesThis fundamental result in real analysis explains…

Answered: THEOREM 11.3.6 A nondecreasing sequence…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

See similar textbooks

Similar questions

Prove that a bounded nonincreasing sequence converges to its greatest lower bound.
3. Prove Bolzano-Weierstrass Theorem. a bounded sequence of real numbers has a convergent subsequence.
Prove that every bounded below, decreasing sequence converges
Prove Theorem 10.2 for bounded decreasing sequences. Theorem10.2 is All bounded monotone sequence converge. please show clear,thanks
B2. (a) Explain in detail what it means to say that a real sequence (an) is bounded. (b) Prove that every convergent sequence is bounded. (c) Prove that the sum of two bounded sequences is bounded. (d) Explain in detail what it means to say that a real sequence (an) diverges to ∞. (e) Suppose that the sequence (an) is bounded and the sequence (bn) diverges to . Prove that the sequence (an + bn) also diverges to ∞.
2. (i) Which of the following statements are true? Construct coun- terexamples for those that are false. (a) sequence. Every bounded sequence (x(n)) nEN C RN has a convergent sub- (b) (c) (d) Every sequence (x(n)) nEN C RN has a convergent subsequence. Every convergent sequence (x(n)) nEN C RN is bounded. Every bounded sequence (x(n)) EN CRN converges. nЄN (e) If a sequence (xn)nEN C RN has a convergent subsequence, then (xn)nEN is convergent. [10 Marks] (ii) Give an example of a sequence (x(n))nEN CR2 which is located on the parabola x2 = x², contains infinitely many different points and converges to the limit x = (2,4). [5 Marks]
7. Does the sequence converge or diverge? If it converges, then find the limit: bn = 3n - √9n² - n
5. Show that the sequence Pn = 1 converges linearly to p = 0, and the sequence Pn = 10-2" converges quadratically to p 0
(Q3) (a) Prove that if (an) and (bn) are sequences such that (an) converges and (an-bn) converges, then (bn) converges. (b) Suppose that (n) is any converging sequence and (yn) is any diverg- ing sequence. Can you say in general whether (n - Yn) converges or diverges? Justify your answer. [Hint: Use the previous part.]
Let a1 = sqrt{2} and ak+1 = sqrt{2+sqrt{ak}}, k = 1,2, ... Show that the sequence {ak} is convergent.

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

THEOREM 11.3.6 A nondecreasing sequence which is bounded above converges to the least upper bound of its range. A nonincreasing sequence which is bounded below converges to the greatest lower bound of its range.