Explain

28. Let (fuof2, .. .) be the sequence of functions with domain I = [0, 1] defined by fn(x) = nx(1 -x)". (i) Show that (f) converges pointwise to the constant func- tion g(x) = 0. (ii) Show that (f,) does not converge uniformly to g(x) = 0. (iii) Show that, in this case, lim S floor ax = S [lim In( ) de 29. Let (fufz, ...) be the sequence in F(R, R) defined by f, (x) = "+1 (i) Show that (f,) converges uniformly on compacta to the function g(x) = x. (ii) Show that (f,) does not converge uniformly to g(x) = x. 30. Let (f1, f2, . . .) be a sequence of (Riemann) integrable functions on / = [0, 1). The sequence (f, ) is said to converge in the mean to the function g if lim S "Ifn (x ) - o(2)12 dx = 0 (i) Show that if (f,) converges uniformly to 9, then (f,) converges in the mean to 9. (Hi) Show, by a counterexample, that convergence in the mean does not necessarily imply pointwise convergence. THE FUNCTION SPACE C[0, 1] 31. Show that ([a, b] is isometric and hence homeomorphic to C[0, 1]. 32. Prove: Let (f,) converge to g in ([0, 1] and let X - 20. Then lim f, (2,) = 9(x,). 33. Let p be a polygonal are in C[0, 1] and let $ > 0. Show that there exists a sawtooth function o with magnitude less than as, i.e. lol)