
5. Give an example of a bounded sequence {sn)1 such that the set con- sisting of...
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. Give an example of a sequence {sn} that is not monotone, but the se- quence {s} is monotone. (7 points) carlo ST 4. Let $i = 4 and 9n+1 = (38m + 1)/5 for n 2 1. Show that the sequence {sn} is bounded and monotone, and find its limit s. (10 points)
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...
2. (5 points) Let {sn}nen be a sequence. Let S be the set of subsequential limits of {Sn}nen, that is, x E S if and only if 3{Sn}ken subsequence of {Sn}nen such that limky Sny = x. Use the previous problem to show that inf S = lim inf sns sup S = lim sup sn.
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
5. Let (In be a nested sequence of closed bounded intervals. For each n E N, let Xn E In. Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals Property
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
6. Give an example of a convergent sequence of numbers n with the property that n I for all n, but for which lim . Prove your claim. Remember to make use of Definition 1.2.
2. (10 Points) Give the following examples (the roofs are not required). (a) A bounded sequence in LP[o 0, 1],1 S p S oo, that has no strongly convergent subsequence (b) A bounded sequence in L'(0, 1] that has no weakly convergent subsequence. (c) A weakly convergent sequence in L [0,1] that has no strongly convergent subsequence.
2. (10 Points) Give the following examples (the roofs are not required). (a) A bounded sequence in LP[o 0, 1],1 S p S...