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.
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Explain the key points of 11.3.6 and prove it
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- 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.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](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.
- 2. Which of the following products are absolute convergent? Determine the corresponding values when they exists. (a) II (¹). k=2 (b) II (1-(-1)) k=1 (c) II (1 k=2 - 2 k3+1 3 (d) II (1-k(k²+2)) k=2Assume that the sequence shown below converges and find its limit. V10, /10 + V10 , /10 + /10+ V10 , 10 + / 10 + V 10 + / 10 , .. The sequence converges to lim an %3D (Type an exact answer.)Determine whether each of the following sequences (an) converges, and find the limit of each convergent sequence. Name any results or rules that you use. You may use the basic null sequences listed in Theorem D7 from Unit D2. (a) an (b) an = (c) an = 4(2") + 3n4 +6 3(2n)+n4+2n³¹ n5 + 3n² + 2n 4n4 + 2n² - n n = 1, 2, ... n = 1, 2, ... n² + 3n² + 2(n!) n³+2n² + (-1)" (n!) n = 1, 2, ...