The sequence 4 xk I} in R" is said to converge to x E R\" Eff for all e :> 0 there exists \"(6) E N such that k 33 N(E) =} d(xk,x) i: E. When answering the following questions, reference any theorem that you appeal to in deriving your answer. Let = 3k+1 k + 3 (a) Use Theorem 1.6 in the lecture notes to derive the limit of as k -> co. Report all steps in the process. No marks will be awarded for an unsupported answer. (b) Is a bounded sequence? Briefly explain. (c) According to Definition 1.11 in the lecture notes, what must you show to prove that L is the limit of , where L is your answer to (a) above? (d) Use Definition 1.11 in the lecture notes to prove that - L, where L is the limit you derived in (a) above. Report all steps in the process. No marks will be awarded for an unsupported answer.Theorem (1.6 continued) (c) X lim XK -,y70. k-+00 "The limit of a ratio of convergent sequences is equal to the ratio of their limits". (d) 1 lim = - , x *0. k -+0o XK X This is a special case of (c) in which YAL = Theorem (1.6) Let and be two sequences in R" and let lim = x and lim = y. k -+ 0o K -+ 0o (a) lim =x_y. K -+ 0o "The limit of a sum (difference) of convergent sequences is equal to the sum (difference) of their limits". (b) lim (XKyK) = xy. k -+0o "The limit of a product of convergent sequences is equal to the product of their limits"