Foundations of analysis
Prove that every finite subset of Rd is closed.
Foundations of analysis Prove that every finite subset of Rd is closed.
Real Math Analysis Let A be a nonempty finite subset of R. Prove that A is compact. Follow the comment and be serious Please. our goal is to show that we can find a finite subcover in A. However, I got stuck in finding the subcover. It is becasue finite subset means the set is bounded but it doesn't mean it is closed.
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.
Let A be a subset of a finite group G with |A| > |G|/2. Prove that every element of G can be written as the product of two elements of A. Is this also always true when |A| = |G|/2?
Prove that a subset of a countably infinite set is finite or countably infinite.
1. Prove that no proper subset of RRR is simultaneously open and closed in the RR topology). 2. Prove that no proper subset of RFC is simultaneously open and closed.
Let 2 [0, 1], and let F be the collection of every subset of such that the subset or its complement is countable. Let P(.) be a measure on F such that for A E F, P(A) if A is countable and P(A)1 if Ac is countable. (a) Is F a field? Also, is F a σ-field? (Note that afield is closed under finite union while a σ-field is closed under countable union. (b) Is P finitely additive? Also, is...
11*. Suppose S a nonempty subset of a group G. (a) Prove that if S is finite and closed under the operation of G then S is a subgroup of G. (b) Give an example where S is closed under the group operation but S is not a subgroup.
1. (a) Prove that a closed subset of a compact set is compact. (b) Let a, b € R and f: R → R, x H ax + b. Prove that f is continuous. Is f uniformly continuous?
Let Ω = [0, 1], and let F be the collection of every subset of Ω such that the subset or its complement is countable. Let P(·) be a measure on F such that forA∈F,P(A)=0ifAiscountableandP(A)=1ifAc iscountable. (a) Is F a field? Also, is F a σ-field? (Note that a field is closed under finite union while a σ-field is closed under countable union.) (b) Is P finitely additive? Also, is P countably additive on F ?
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...