(6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S.
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.
#3 A Supremely Infimum Problem (Zorn 1.9 #8) Let S R be non-empty and bounded below. Let-S f-xlxES). Show that sup(-S) exists. Then show that -inf (S) sup(-S). This problem shows that the completeness axiom guaranteeing the existence of supremums implies a similar statement about the existence of infimums. Write down an "infimum" version of the completeness axiom. that-1 #3 A Supremely Infimum Problem (Zorn 1.9 #8) Let S R be non-empty and bounded below. Let-S f-xlxES). Show that sup(-S)...
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.
Let A, B be non-empty, bounded subsets of R. a) If the statement is true, prove it. If the statement is false, give a counterexample: sup(AUB) = max(sup(A), sup(B)}. b) If the statement is true, prove it. If the statement is false, give a counterexample: If An B + Ø, then sup(A n B) = min{sup(A), sup(B)}. E 选择文件
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...
2. Let A be a non-empty subset of R bounded below. Show that inf (A) is a border point of 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.
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
Problem 4 [20] Suppose A C R is a non-empty set, and sup(A), inf (A) exist. Show that sup(A) inf (A) if and only if |A. Problem 5 [25]
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...