3. Give an example of a sequence {sn} that is not monotone, but the se- quence...
Separate each answer? 5) Define the supremum of a bounded above set SCR. 6) Define the infimum of a bounded below set SCR. 7) Give the completeness property of R 8) Give the Archimedean property of R. 9) Define a density set of R. 10) Define the convergence of a sequence of R and its limit. 11) State the Squeeze theorem for the convergent sequence. 12) Give the definition of increasing sequence, decreasing sequence, monotone se- quence. 13) Give the...
Please answer all parts. (2) (a) Give an example of sequences (sn) and (tn) such that lim sn ntoo 0, but the sequence (sntn) does not converge does not converge.) (b) Let (sn) and (tn) be sequences such that lim sn (Prove that it O and (tn пH00 is a bounded sequence. Show that (sntn) must converge to 0. 1 increasing subsequence of it (b) Find a decreasing subsequence of it (3) Consider the sequence an COS (а) Find an...
3. Let (an)n1 be a sequence. o Prove that if (an)ni is monotone increasing and not bounded above, thenlimn00 an0o. o Show that removing the monotonicity hypothesis makes this statement false. (Give an example of a sequence that is not bounded above, and does not diverge to oo.)
Question 2. Monotone Convergence Define a sequence (an) inductively by ai = 1 and an+1 = ("p) (a) Show that, for any k E N, if 0 <a << 2 then 0 < ak+1 <2, and deduce that a, E (0,2) for all E N (b) Show that the sequence (an) is increasing and bounded above. (c) Prove that the sequence converges, and find its limit Question 2. Monotone Convergence Define a sequence (an) inductively by ai = 1 and...
5. Give an example of a bounded sequence {sn)1 such that the set con- sisting of all its subsequential limits is precisely the closed interval [0, 1]. (Prove that your example has this property).
4. Use the Monotone Convergent Theorem (Theorem 4.3.3) to prove that the following sequence is convergent, then find its limit. (Hint: You will need mathematical induction). S1 = 1 and Sn+1 = (2 sn + 5) forn EN
Show that the sequence is Cauchy using the definition of Cauchy se- quences. Sn 2n +1 n +4
(9 marks) Let { ln(n+11) n+3 }n=1 be a sequence. a. Find the first 5 terms of the sequence in the exact form. b. Determine whether the sequence is strictly monotone, monotone, eventually strictly monotone, eventually monotone or neither. Prove it. c. Determine whether the sequence is convergent, and if so, find its limit.
show all work | 2n-1) 2. Consider the sequence |(n+1)! a) is the sequence monotone increasing or monotone decreasing or neither? b) Find upper and lower bounds for the sequence. c) Does the sequence converge or diverge? (Explain) 3. Determine if the series converges or diverges. If it converges, find its sum. => [-1-] c) Ë ?j? – 1-1 j? +1