Problem 3: Let the set of rational numbers in [0, 1] be denoted by Q = {r1, r2 , . .., rn , . .. }, and define fn(2) =0, if x E {r1, r2, ..., Tn}; fn() =1, if x & {ri,r2, ..., In}. (i) Show that fn is Riemann integrable over [0, 1]. (ii) Show that fn - f (the Dirichlet function as defined in Problem 1) everywhere in [0, 1]. (iii) Study whether the limit fn -> f in (ii) is uniform over [0, 1]. (Hint: Use Problem 2). Problem 4: Study whether the bounded convergence theorem is valid for the Riemann integral. (Hint: Consider Problem 3.) Problem 5: For the sequence of the functions {fn} defined in Problem 3, define gn = 1 - fn. (i) Show that {on} is a monotone increasing and nonnegative sequence that converges to g = 1 - f everywhere. (ii) Use this sequence to study whether the monotone convergence theorem is valid for the Riemann integral