Could someone please check my work

Suppose that (S, ) is a convergent sequence with a S s., 0, 3N E N such that Vn > N, Is, - s| NV by Theorem 3.2.10(b). To prove that a _ lim 8, = s : Suppose that s Sn 0, 3N E N such that Vn > N, Isn - s > - & for all n 2 NV by Theorem 3.2. 10(b). Suppose that s > b , so that the limit of Sn is greater than the supremum, b , of Sn . Since b E (Sn), then by the previously stated definition of convergence, b - s| s - & by Theorem 3.2.10(b), which means b - s > -E and by Theorem 3.2.2 (f ) , s - b b , then 0 b - s Since Sn > S - E and - E > b - s, Sn > 8 - E > s+ ( 6 - s) = b which means Sn > b , which is a contradiction. Thus, b 2 lim Sn = $ Since a lim Sn = s , = a