11. [8 marks] Let / E R. Suppose that {an} converges to 4. Provide a complete and accurate e - N proof that {a? } converges to (2. BONUS [8 marks] Let I = Jo (vr + 1 - cos(2x)) da. Find the smallest n value that will guarantee Simpson's rule S, approximates / with error strictly less than 0.001. 3. Let {an } be a sequence defined recursively by a1 = V6 anti = V6 + 5an, n = 1,2,3, ... (a) [9 marks] By induction or otherwise, show that { an } is increasing and bounded above by 6. 4. [11 marks] Let b, c, (, m E R. Let f, g: R - R be functions. Suppose that lim f(x) = ( and lim g(x) = m. Give a complete and accurate e - N proof that lim (2bf(x) - 3cg(x)) exists. 5. [9 marks] Let function g: R - R be defined by g(x) = 6 ,ifreQ 1 , ifreQ Prove that g is not integrable on [0, 1]