Could someone please check my work

1. If (Sn) is an unbounded increasing sequence, then lim Sn = + co. Let (Sn) be an increasing sequence and suppose that the set S = {Sn : n E N) is unbounded. Since (Sn) is a sequence by definition 4.4.1 which is increasing by definition 4.3.1, then S is bounded below by $1 . Hence, S must be unbounded above. Thus, given any M E R , there exists a natural number N such that SN > M by the definition of divergence. But then for any n _ N we have Sn _ SN > M , so lim Sn = + co by the definition of divergence. 2. If ( Sn) is an unbounded decreasing sequence, then lim Sn = - co. Let (Sn) be a decreasing sequence and suppose that the set S = {sn : n E N} is unbounded. Since ( Sn) is a sequence by definition 4.4.1 which is decreasing by definition 4.3.1, then S is bounded above by $1 . Hence S must be unbounded below. Thus, given any M E R , there exists a natural number NV such that SN