1. Prove that any infinite set contains a countable subset (see Problem 20, page 43)
Prove that every subset of N is either finite or countable. (Hint: use the ordering of N.) Conclude from this that there is no infinite set with cardinality less than that of N.
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection between A and some infinite subset of N.) Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection...
Prove that a subset of a countably infinite set is finite or countably infinite.
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....
Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X is closed and bounded but NOT compact
Problem 1. Let A C R be a countable set. Prove that R\ A is uncountable.
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the set of all real numbers IR is measurable with >(IR) = . 8) Prove that If f : [a, b] IR is continuous [a; b]then it is measurable [a, b]. 9) Give an example of a function f : [O, 1] IR which is measurable on [O, 1] but not continuos on [O, 1]. 10) Find the Lebesgue integral...
4. Show that a set A is infinite if and only if it is equivalent to a proper subset of itself (Problem 21 on page 43).
Problem 6 Suppose A and B are countable sets. Prove A × B is a countable set.