14) [12 Marks, 4 marks for each part] Closed and Compact Sets. A set F C R is said to be closed if whenever {, } is a sequence in F which converges to some Fo, then To E F. A set K is said to be compact if whenever {r, } is a sequence in K, there is a subsequence {In } which converges to some fo E K. a) Show that if K C R is both closed and bounded, then K is compact. b) Show that if K C R is compact, then K is both closed and bounded. c) [Extreme Value Theorem:] We say that a function f is continuous on a set S C R if whenever {{ } is a sequence in S which converges to some ro E S, we have that {f(In) } converges to f(Io). By modifying the proof of the Extreme Value Theorem show that if K is compact and non- empty, and if f is continuous on K, then there exists a ce K such that f(x)