No Contradiction 2. Let A and B be non-empty subsets of R, and suppose that ACB....
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 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 : 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.
2. Let A be a non-empty subset of R bounded below. Show that inf (A) is a border point of A
Let T be a bounded subset of R and let S CT. Prove that supS < supT.
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.
Let A, B, C be subsets of U. Prove that If C – B=0 then AN (BUC) < ((A-C)) UB
1. Let x, a € R. Prove that if a <a, then -a < x <a.
2, For each of these sets. A={3n : n E N), B = {r E R : x2 < 7), and C = {x E R : x < 12), (i) Is the set bounded above? Prove your answer.] ( .] ii) Is the set bounded below? Prove your answer answer the following questions:
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0