2 T/ N~ COS (W) (b.) Next, consider the sequence b, = i1 (i.) Find an infinite sequence of m values such that b, = 0. ii.) Prove, using an N proof, that the lim b, # 0. g noo [Hint: Choose an appropriate and then show that, no matter what N is chosen, there will always be a n > N such that |b, 0| > ] (iii.) Prove, using an N proof, that the li_>m b, # K, for any K # 0. noo [Hint: Choose an , in terms of K, and then show that, no matter what N is chosen, there will always be a n > N such that |b, K| > ] In this problem, we will use e - N arguments to determine the convergence of some sequences a. Consider sequence a = 1/1 + i. Come up with a conjecture (answer w/o proof) for the value of lim a at several n values. Call this limit L. When n = 10, a = V/1+4= 10 = a = v1. 1 When n = 100, a =/1 + 100 -= a = v1. 01 When n = 1000, a = 1 + 1000 - = a = v1. 001 Therefore, as n gets infinitely big, the limit goes to 1. So, L = 1. ii. Let E = 0. 1 Find the smallest N such that for every n 2 N, la - L| 5 Given E = 0. 01, la - L| 50 Given E = 0. 001, la - L| 0 Given e, set 1 + e=1 + - and solve for n. Round up to the next natural number to use as N