I'm lost with this. Please help me out.

Let 01 ([0, 1], R) denote the collection of all the realvalued, continuously differentiable functions on [0,1]. In other words, 01([0,1],R) = {f : [0,1] > R | f and f' are both continuous on [0,1]}. (1) Note that this is a subset of C ([0, 1], R). Dene the sup norm in 01([0, 1], R) by ||f||01= sup |f($)|+ sup |f'($)l- (2) $E[0,1] :66 [0,1] The natural metric induced by the sup norm is d( f, g) = H f g||g1. Let A be the closed unit ball in 01([0,1],R), i.e., A = {f E 01([0,1],R) | ||f||01 5 1}. (a) Show that A is compact in C([0,1],R). (b) Show that A is not compact in 01([0, 1], R). (Hint: consider the sequence fn(:r) =