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)

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.

Problem 2. Let f : R → R be a function. Suppose that f is increasing: for any x, y ∈ R,if x ≤ y then f (x) ≤ f (y). Prove that limx→a− f (x) exists for any a ∈ R. (Hint: What shouldthe value of this limit be? Try sup{f (x) : x < a}.)

Let X and Y be non-empty sets, and let h : X × Y → R be a bounded function.(i) Prove thatsupy∈Ysupx∈Xh(x, y) = sup(x,y)∈X×Yh(x, y) = supx∈Xsupy∈Yh(x, y).Hint) Interchanging the roles of X and Y if necessary, it is enough to prove thatsupy∈Ysupx∈Xh(x, y) = sup(x,y)∈X×Yh(x, y)

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

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