Solved using basics of real analysis
Let A be a nonempł4 subset of IR that is bounded below and let x =...
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
APPLICATIONS OF THE COMPLETENESS AXIOM 1.5.5 Let A be a nonempty subset of R. Define -A={-a: a E A}. (a) Prove that if A is bounded below, then -A is bounded above. (b) Prove that if A is bounded below, then A has an infimum in R and infA = - sup(-A).
Let T be a bounded subset of R and let S CT. Prove that supS < supT.
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...
2. Let A be a non-empty subset of R bounded below. Show that inf (A) is a border point of A
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,...
Let A be a nonempty subset of R. Define -A={-a: a A}. (a) Prove that if A is bounded below, then -A is bounded above. (b) Prove that if A is bounded below, then A has an infimum in R and inf A=-sup (-A).
Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X is closed and bounded but NOT compact
9. (10 points) Let R be a ring and let X be a subset of R. De X Prove that A(X) is a subring of R and give an example to show that A Ir e R: r be an ideal in R. x) need not
please provide detail! will rate! thank you! 4. Let C be a closed, and bounded subset of IR". Suppose that 01,02, Os, is a sequence of open subsets of Rn and C u 10k. Prove that there exists m E N such that C ur10k. Here is a hint. First of all, for m e N, et nO We have ui S tus s c ume-iu,n You are given that cach Oh in open what can you say about u....