6. Let X be a non-empty subset of an ordered field with the least upper bound...
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...
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
Prove that the real numbers have the least upper bound property, i.e. any bounded above subset S ⊆ R has a supremum if and only if the real numbers have the greatest lower bound property, i.e. any bounded below subset T ⊆ R has an infimum.
Prove: Let A be a dense subset of (X, T), and let B be a non-empty open subset of X. Then An B Prove: Let A be a dense subset of (X, T), and let B be a non-empty open subset of X. Then An B
2. Let A be a non-empty subset of R bounded below. Show that inf (A) is a border 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 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).
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.
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.
Question 2. Prove that if S C R is bounded above then its least upper bound is unique. Le, that if X,, R are both least upper bounds for S then ג ,