Could someone please check my work

Find The limit of The following sequence and determine if The Sequence Converges or diverges. S. = not +n By Theorem 4.2.13, S. diverges to too if lime ( 5 ) = 0 To Show lim (2 ) =n'tn 140 - = 0, apply The definition of convergence. Given EXO, Take NEN such That N > max /8, ES . Then n2N implies That no8 and ME. Since no 8, we have nath sant and 4" 2 n . Thus for n > N, we have 1 +n 4 +n 2n 2 4n 3 Hence, by definition of convergence and Theorem 3. 3. 10 ( c ) for The Archimedean property , lim n tn = 0 . 4 0 Since like n 4 0 4 0 = 0, Then by Theorems 4.2.13, S, - diverges to too