Please solve these questions related to Real analysis course. I really need to make sure of my own answers. 3. Let f : [2, 3] {1,3} be a continuous function such that the equation f(x) – a = 0 has a solution ain [2, 3]. What can you say about the function f?4. Let A be a set of real numbers such thatinf{|a1 – az| : a1, az e A a1 + az} > 0.Prove that every Cauchy sequence of elements in A converges to a point of A.5. Let ceR and suppose f, g: R -→ R are functions such that lim,e f (x) exists, and for some e > 0,|9(x)| < |f(x)| whenever |2 – c| < e. Explain whylim (sin (x – c)g(x)]exists. State explicitly any theorem or result that you are using in your argument.6. Let f, g : R → R be continuous functions and defineS = {r €R: f(x) – g(x) € Z},where Z stands for the set of all integers. Let (r,) be a convergent sequence in S and let x =lim,+ *n. Can we say that r € S? Would your answer change if we replace Z with the set Q of allrational numbers? Justify your answer.

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3. Let f [2,3] {1,3} be a continuous function such that the equation f(x) - r = 0 has a solution a in [2,3]. What can you say about the function f? 4. Let A be a set of real numbers such that infla1-a2a1, a2 € A a1 + a2} > 0. Prove that every Cauchy sequence of elements in A converges to a point of A. 5. Let e ER and suppose f.g: RR are functions such that lim, f(x) exists, and for some > 0, lg(x) ≤ f(x)| whenever r- c

3. Let f [2,3] {1,3} be a continuous function such that the equation f(x) - r = 0 has a solution a in [2,3]. What can you say about the function f? 4. Let A be a set of real numbers such that infla1-a2a1, a2 € A a1 + a2} > 0. Prove that every Cauchy sequence of elements in A converges to a point of A. 5. Let e ER and suppose f.g: RR are functions such that lim, f(x) exists, and for some > 0, lg(x) ≤ f(x)| whenever r- c

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 48E: Let R be the set of all infinite sequences of real numbers, with the operations...

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