PROBLEM 5. Let S be a nonempty subsets of R. Suppose f:SR and g:SR are bounded functions (this means their ranges are bounded subsets of R. (a) Show that {f(x)+g(x):xS}{f(x):xS}+{g(x):xS} (b) Prove that sup{f(x)+g(x):xS}sup{f(x):xS}+sup{g(x):xS}. (c) Give an example of two functions for which the inclusion in part (a) is strict. (d) Show that the inequality in part (b) can be strict