1-5 . use the completeness axiom to prove that every nonempty set of real numbers E that is bounded below has an infimum and that Inf E= sup {-x: |x∈E} Leave a Comment / Calculus, Mathematics / By wegoki EXPERT ANSWER

3. (10 points) carefully state the axiom of completeness and use it to prove that every bounded increasing sequence of real numbers has a limit Leave a Comment / Calculus, Mathematics / By wegoki EXPERT ANSWER

State the completeness axiom for extended reals(R*). Give an example to show that the set of all rational numbers is not complete. Leave a Comment / Calculus, Mathematics / By wegoki EXPERT ANSWER

2a) state the axiom of completeness for the set R of real numbers. Leave a Comment / Calculus, Mathematics / By wegoki EXPERT ANSWER

Problem 4. Prove using the completeness property that √2 is a real number by considering the set {x∈(0,2)|x^2<2}. Leave a Comment / Calculus, Mathematics / By wegoki EXPERT ANSWER

Use the Completeness Axiom to prove that every nonempty set of real numbers that is bounded below has an infimum. Leave a Comment / Calculus, Mathematics / By wegoki EXPERT ANSWER

A) State the completeness Axiom for real numbers, prove that the infimum of the interval (3,7) equal 3. Leave a Comment / Calculus, Mathematics / By wegoki ANSWER

Infimum version of the Axiom of Completeness. Assuming the Axiom of Completeness, prove that any nonempty set of real numbers A which is bounded below has a greatest lower bound. Hint. Consider the set {−a : a ∈ A}. Leave a Comment / Calculus, Mathematics / By wegoki solution

Problem 5. Show that the assertion of the Heine-Borel theorem is equivalent to the completeness Axiom for the real numbers. Leave a Comment / Calculus, Mathematics / By wegoki solution

Question. Open exercise 2.18 let (a_n) be an increasing bounded sequence.by the completeness Axiom the sequence converges to some real numbers a. show that its range {a_n| n∈N} has a supremum and that the supremum equals a Leave a Comment / Calculus, Mathematics / By wegoki solution