11) Suppose that an 2 0 and that lim an = L. We will show that n -too lim Van = VL . n -+oo There are two cases to consider: a) Suppose that L = 0. Let e > 0. Choose No E N so that if n 2 No (Choose 1) i) then 0 No, then I van - 0 |= Van 0. Let e > 0. Note that I van - VI |= Jan - L| |an - LI Van + VL- VI Choose No large enough so that if n 2 No (Choose 1) i) then | an - L KE. ii) then | an - L K VL . E. iii) then | an - L K V. It follows that if n > No, then I van - VIKE. Hence, lim Van = VL. c) Let an = 4 + 2. Note that an - 4. By making use of what we saw in b) above find a cutoff No so that if n 2 No, 1 1/ 4+2 - 2K. 1 1000 d) Let a1 = 1 and let anti = V3 + 2an. {an} is both monotonic and bounded. Find lim an