#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)...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
5a) (5 pts) Find lim inf (xn) and lim sup (rn), for rn = 4 + (-1)" (1 - 2). Justify your answer 5b) (5 pts) Find a sequence r, with lim sup (xn) = 3 and lim inf (x,) = -2. 5c) (10 pts) Let {x,} be a bounded sequence of real numbers with lim inf (x,) = x and lim sup (x,) = y where , yER. Show that {xn} has subsequences {an} and {bn}, such that an...
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.
Let (, A, ) be a measure space. Let fn : 2 -» R* be a sequence of measurable functions. Let g,h : 2 -» R* be a pair of measurable functions such that both are integrable on that a set A E A and g(r) fn(x)h(x), for all E A and nE N. Prove / fn du lim sup A lim inf fn dulim inf lim sup fn du A fn du no0 no0 A noo n+o0 (You may...
Please help me solve 3,4,5 3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.
Let the function f: (a, b) → R is continuous in (a, b). If sup {f(x): x ∈ (a, b)} = L> 0 and inf {f(x): x ∈ (a, b)} = M <0, then prove that there is a c ∈ (a , b) such that f (c) = 0.
8 arbitrary set. K is Cousider E} n=1 nieU and Let (X, K) be a measure space where X is an sigma-algebra of subsets of X and is a measure sequenc o clemenis of K We delin lim supn(Fn) liminfn(En)- U then prove: (a) lim in(E)) lim inf(u(E,) (b) T J (c) If sum E,)x, then (lim sup(E)) = 0 x X) <oc lor somc nE N, then lim supn (Fn)> lim sup(u(F,n )) 8 arbitrary set. K is Cousider...
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,...