Find the infimum and supremum, if they exist, of each of the following sets.(Indicate that the infimum or supremum does not exist if they do not exist. Justificationsare not required.)A := {2n : n ∈ N},B := {2−n : n ∈ N},C := 12n + 1 : n ∈ N

Problem 6. Let A, B ⊆ R be nonempty and bounded sets.(a) Prove that if A ⊆ B, then inf B ≤ inf A ≤ sup A ≤ sup B.(b) Prove that sup(A ∪ B) = max{sup A, sup B}. (Hint: A is a subset of A ∪ B, so we canapply part (a) to these two sets.

LetA :=32 − (−1)n2n : n ∈ N.Find inf A and sup A. Justifications are required

List the elements of the setsC = {(n + 1)2 − (n2 + 2n) | n ∈ N}

Let S1, S2, S3, . . . be an infinite collection of non-empty sets of real numbers,and let S be their union:S =∞[i=1Si.(a) Suppose that S has a supremum. Explain briefly why each Si has asupremum. [3 marks](b) If each of the sets Si has a supremum, must S also have a supremum? Ifyes, give a proof. If no, give a counterexample and justify it

et f : R → R be a function, and consider the two setsX = {f (x)2 : x ∈ R}, Y = {f (x2) : x ∈ R}.Show that if X has a supremum, then Y has both a supremum and an infimum.Give a counterexample to show that the converse is false (that is, Y can havea supremum and infimum, even if X has no supremum)