100 n?+1 n? cos (\""ln) (b.) Next, consider the sequence b, = * (i.) Find an infinite sequence of m values such that b, = 0. (ii.) Prove, using an N proof, that the lim b, # 0. =00 [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 nl'i}rrr}obn # K, for any K # 0. [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 + 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 =1+ --= a =v1.1 When n = 100, a =q1+7=a = v1.01 When n = 1000, a = 1 + 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| 0 Given E, set 1 + e =/1 + - and solve for n. Round up to the next natural number to use as N