Math 74 Homework 14Due Monday, December 1stNovember 24, 20081. (a) Show that R2(= R × R) is Cauchy complete with respect to theEuclidean metric.(b) Show that the set {(x, y) ∈ R2| 1 ≤px2+ y2≤ 2} is Cauchycomplete with respect to the Euclidean metric.2. Let (X, d) be a metric space and let Y ⊆ X be a subset. Show thatx ∈ X is a limit point of Y if and only if there exists a sequence (yn)of elements of Y such that (yn) converges to X.3. Let (X, d) be a metric space and let Y ⊆ X be a subset. Define theclosure¯Y of Y to be the intersection of all closed subsets of X whichcontain Y .(a) Show that¯Y is the smallest closed subset of X containing Y , i.e.show that¯Y is closed, that Y ⊆¯Y , and that if Z ⊆ X is anyclosed subset of X such that Y ⊆ Z, then¯Y ⊆ Z.(b) Show that the property in (a) totally determines¯Y , i.e. showthat if C ⊆ X is a closed set such that Y ⊆ C and such that C iscontained in any other closed Z which contains Y , then C =¯Y .(c) Show that¯Y = {x ∈ X | x is a limit point of Y }.4. As in the previous exercise, let (X , d) be a metric space and let Y ⊆ Xbe a subset. Define the interior Y◦of Y to be the largest open subsetof Y . Show that Y◦exists and is unique, give a nice description of Y◦,and relate Y◦to limit points in X.5. Let (X, d) be a metric space, and let Y and Z be two nonempty subsetsof X. Define the distance from Y to Z, dist(Y, Z), to be the greatest1lower bound of set {d(y, z) | y ∈ Y, z ∈ Z}. Explain why dist(Y, Z) ex-ists. For each of the following statements, either prove the statement,or give a counterexample (very convincing pictures are acceptable):(a) For any three nonempty subsets W, Y, Z ⊆ X,dist(Y, Z) ≤ dist(Y, W ) + dist(W, Z).(b) If dist(Y, Z) = 0 then Y ∩ Z 6= ∅.(c) If Y and Z are closed subsets of X and dist(Y, Z) = 0 thenY ∩ Z 6=