Question 2. Prove that if S C R is bounded above then its least upper bound...
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.
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...
7. Let E C R be nonempty, n E N, and K, L E Z such that K/n is an upper bound for E, but L/n is not an upper bound for E. (a) Show that there exists an for E, but (m - 1)/n is not an upper bound for E. (Hint: Prove by contradiction, and use induction. Drawing a picture might help) m < K such that m/n is an upper bound integer L (b) Show that m...
4. For the following sets determine the least upper bound (it is not necessary to prove that it is the least upper bound): a.) M = [0; 1] [ (3; 4) b.) M = n5n + 1 4n ? 3 n 2 N o c.) M = n n + 1 2n + 13 n 2 N o d.) M = nXn i=1 9 10i n 2 N o e.) M = n xjx > 0 and x2 < 5g:...
6. Let X be a non-empty subset of an ordered field with the least upper bound property. Supposed that X is bounded above and define -X = {-1 : TEX} Prove that supX = - inf(-X).
Prove this theorem.. Suppose S is a set of numbers, M is the least upper bound of S and M does not belong to S. Then, M is a cluster point of S.
QUESTION 7 Consider the poset (A, R) represented by the following Hasse diagram (2 (a) Give each of the following If any do not exıst, explan why (i) The greatest element of (A, R) (i:) The least element of (A, R) (i) All upper bounds of {h, eh (iv) The least upper bound (LUB) or(h (v) All lower bounds of (b,c) (vi) The greatest lower bound (GLB) or(b, c} (b) Give complete reasons for the answers to the following (i)...
4.1. Let S, TCR. Prove that (i). SCT S'CT. 4.2. Let S, TC R. Prove that (i. SCT SCT. 4.3. Show that if the set S is bounded above or below, then so is S' with the bounds of S and has greatest or smallest member accordingly, provided S' 0.
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.
please answer both questions Let ג: ] 9,00)-C be a continuous and bounded function such that the limit d exists f(t) dt = Let ริ่ be the LapIde transform oss, î.e.. Prove that km s1(s) = d 〈this is a version of the final vahe theorem Give an exa mple of f as in 3 which does not have a limit as t-> oo Let ג: ] 9,00)-C be a continuous and bounded function such that the limit d exists...