Which of the following statements are true for the sequence: ' ( ) 2 Select all true options The sequence has a convergent subsequence The sequence has a divergent subsequence The sequence converges The sequence diverges The sequence has two convergent subsequences with different Emits Choose all correctly presented solutions You can select multiple choices Suppose that a sequence a,, satisfies a, 0if x 2 1. hence both f (x) and Cn are non-decreasing. Let Cn = e" - n. Let f(x) = e" - x. Since f'(x) = ex - 1 > 0if x 2 1, hence both f (x) and cn are non-decreasing. Therefore, by the Monotone Convergence Theorem, lim C,, exists. 1-+00Which of the following reasons or workings are correctly stated? Select all correct options You can select multiple choices lim n-+Do (1 + 1 ) = lim 1 + lim = = 1+0 =1 lim ( 1+ lim =1' =1 1-+00 lim 1+ = lim 1" = lim 1 = 1 n=100 n-+00 lim (vn + 1 - vn) = lim vn + 1 - lim vn =co - co = 0 1-+00 1-+00 n limn- 180 7 lim = 1 noon + 1 lim, too n + 1 OO