We need at least 10 more requests to produce the answer.
0 / 10 have requested this problem solution
The more requests, the faster the answer.
Let A, B be non-empty, bounded subsets of R. a) If the statement is true, prove...
No Contradiction 2. Let A and B be non-empty subsets of R, and suppose that ACB. Prove that if B is bounded below then inf B <inf A.
Let A and B be two non-empty bounded sets, and A and B are disjoint. Is sup(A U B) = sup(A) + sup(B)? Prove if true, and give a counter example if not.
4·Let A and B be non-empty subsets of a space X. Prove that A U B is disconnected if A n B)U(A nB) 0. Prove that X is connected if and only if for every pair of non-empty subsets A and B of X such that X A U B we have (A B)U (An B)O.
(6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S. (6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S.
REAL ANALYSIS Question 1 (1.1) Let A be a subset of R which is bounded above. Show that Sup A E A. (1.2) Let S be a subset of a metric space X. Prove that a subset T of S is closed in S if and only if T = SA K for some K which is closed in K. (1.3) Let A and B be two subsets of a metric space X. Recall that A°, the interior of A,...
Let A be a non empty subset of R that is bounded below and let a=inf A. If a a&A, prove thal x is a limit point of A
5. Let S be a non empty bounded subset of R. If a > 0, show that sup (as) = a sup S where as = {as : ES}. Let c = sup S, show ac = sup (aS). This is done by showing (a) ac is an upper bound of aS. (b) If y is another upper bound of as then ac S7 Both are done using definitions and the fact that c=sup S.
Let A be a non-empty subset of R that is bounded above. (a) Let U = {x ∈ R : x is an upper bound for A}, the set of all upper bounds for A. Prove that there exists a u ∈ R such that U = [u, ∞). (b) Prove that for all ε > 0 there exists an x ∈ A such that u − ε < x ≤ u. This u is one shown to exist in...
5. Let A and B be compact subsets of R. (a) Prove that AnB is compact (b) Prove that AUB is compact. (c) Find an infinite family An of compact sets for which UAn is not compact. o-f (d) Suppose that An is a compact set for n 21. Prove that An is compact.
5. Let S be a non-empty bounded subset of R. If a > 0, show that sup (aS) = a sup S where aS = {as : s E S}. Let c = sup S, show ac = sup (aS). This is done by showing: (a) ac is an upper bound of aS. (b) If y is another upper bound of aS then ac < 7. Both are done using definitions and the fact that c=sup S.