Please help me answer this question.

Let B = {f E C([0,a]) : |f{t) ml 5 b for all t E [0,6]}. Show that B is e eleeed subset of can, a]), and is therefore e complete metric space. Specifically, we want to show the existence and uniqueness of a solution to the initial value problem y = $(I,y), y(0) = yo- (8 Here, o is assumed to be a continuous and bounded function on the strip R = 0 0, with respect to its second variable: lo(x, y) - 6(x, z)| 0. (We will place additional restrictions on a as the proof proceeds.) Let C([0, a]) denote the metric space of continuous, real-valued functions on [0, a], with the usual metric of uniform convergence: d(f, g) = llf - gll = sup{If (t) - g(t)| : te [0, a]}