5. Let (In be a nested sequence of closed bounded intervals. For each n E N,...
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...
8. Show that Theorem 3.1, the Nested intervals theorem, may be proved as a direct consequence of the Cauchy criterion for convergence (Theorem 3.14). (Hint: Suppose I. = {x: 0, <x<bn} is a nested sequence. Then show that {an} and {b} are Cauchy sequences. Hence they each tend to a limit. Since b.-4, 0, the limits must be the same. Finally, the Sandwiching theorem shows that the limit is in every 1.] Definition. An infinite sequence {n} is called a...
8) This is essentially p.221, #15a), but using more clarified notation. Let D be a closed, bounded interval and f : D → R. Suppose that for each c E D there exists δ = and M = Mc both depending on c where If(x)| < M if |x-c| < δ and x E D. Prove that in fact f is bounded on D. That is, there exists M>0 with If (x)S M for all x E D. Also, find...
Let K1 ⊃ K2 ⊃ K3 ⊃ ... be a sequence of bounded closed sets. Let (an) be a sequence of numbers with the property an ∈ Kn \ Kn+1. Show that (an) has a subsequence that converges to a point a ∈ ??∩Kn. Carefully state which theorems you are using.
13 14 Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
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...
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
2. Let {xn}nEN be a sequence in R converging to x 0. Show that the sequence R. Assume that x 0 and for each n є N, xn converges to 1. 3. Let A C R". Say that x E Rn is a limit point of A if every open ball around x contains a point y x such that y E A. Let K c Rn be a set such that every infinite subset of K has a limit...
If true, prove. If false provide counterexample. Let In be a sequence of nested unbounded open intervals. Then In 0. n=1
Let (an)nen be a bounded sequence in R. For all n e N define bn = sup{am, On+1, On+2,...}. (You do not have to show that the supremum exists.) (a) Prove that the sequence (bn)nen is a monotone sequence. (b) Prove that the sequence (bn)nen is convergent. (c) Prove or disprove: lim an = lim bre. 100 000