Exercise 1.1.2 Let S be an ordered set. Let A CS be a nonempty finite subset....
Let A be a nonempty subset of R. Define -A={-a: a A}. (a) Prove that if A is bounded below, then -A is bounded above. (b) Prove that if A is bounded below, then A has an infimum in R and inf A=-sup (-A).
Real Math Analysis Let A be a nonempty finite subset of R. Prove that A is compact. Follow the comment and be serious Please. our goal is to show that we can find a finite subcover in A. However, I got stuck in finding the subcover. It is becasue finite subset means the set is bounded but it doesn't mean it is closed.
Let A be a nonempty finite subset of R. Use mathematical induction on the number of members of A to show that A has both a largest and a smallest member.
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
2. Let A be a nonempty set of real numbers bounded above. Define Prove that -A is bounded below, and that inf(-A) = -sup(A). -A={-a: aEA . (5 marks) (You may use results proved in class.) A = 0 , A is bounded above.
2. If S:= {1/n - 1/min, me N}, find inf S and sup S. 4. Let S be a nonempty bounded set in R. (a) Let a > 0, and let aS := {as : S ES). Prove that inf(as) = a infs, sup(as) = a sup S. (b) Let b <0 and let b = {bs : S € S}. Prove that inf(bs) = b supS, sup(bs) = b inf S. 6. Let X be a nonempty set and...
APPLICATIONS OF THE COMPLETENESS AXIOM 1.5.5 Let A be a nonempty subset of R. Define -A={-a: a E A}. (a) Prove that if A is bounded below, then -A is bounded above. (b) Prove that if A is bounded below, then A has an infimum in R and infA = - sup(-A).
Problem 1 [10 points] Prove that if A is a nonempty set of real numbers with a lower bound and B is a nonempty subset of A, then inf A <inf B. Problem 2 [10 points) Let A be a nonempty set of real numbers with a lower bound. Prove there exists a sequence (ar) =1 such that are A for all n and we have limntan = inf A.
. (15 points) Let G be a group and A be a nonempty subset of G. Consider the set Co(A) = {9 € G gag- = a for all a € A}. (a) Compute Cs, ({€, (123), (132)}), where e is the identity permutation. (b) Show that CG(A) is a subgroup of G. (c) Let H be a subgroup of G. Show that H is a subgroup of Ca(H) if and only if H is abelian.
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...