Need solution for these four exercises. These questionnaire related to topology and analysis.

Exercise 1 Let A C R be a nonempty and bounded subset of IR. Prove that. sup A e I. Here, IR is equipped with its usual distance. Exercise 2 Let (X ,a'.) be a metric space and let A C X be a nonempty subset. 1. Prove that if 0 is an open set satisfying A C 0 C A then 0 = A. In other words, A is the largest open set contained in A. 2. Prove that if F is a closed set satisfying A C F C K then F = Z. In other words, 3 is the smallest closed set containing A. Exercise 3 Let A and B be two nonempty subsets of IR. 1. Prove that A U B = EU E (Hint: use exercise 3). 2. Prove that A H B C 2 H F (Hint: use exercise 3). 3. Find two such sets satisfying the strict inclusion A H B C E H E. Exercise 4 Determine the limits points, the interior, the closure and the boundary of the following sets. 1. In ]R with the usual distance: {0, 1}, [0, 1) U {2}, Z. 2. In (R2,d2): {(3,310 E R2| - 1