Could someone please check my work

If ( Sn) is on unbounded increasing sequence, Then lim s= to0. Ler (sa) be an increasing sequence and suppose That The Ser S = ESnineIN] is unbounded. Since ( Sn) is increasing, S is bounded below by s,. Hence S Must be unbounded above. Thus, given any MEIR, There exists a natural number N Such That Sy > M by the definition ? of divergence . But Then for any AZ N we have S, 25,, > M, so lim s= too by The definition of divergence. If (s,)is an unbounded decreasing Sequence, Then lim S. =- 0. Let ( sn) be a decreasing sequence and suppose that The ser S= ESni NEIN3 is unbounded. Since ( SA) is decreasing S is bounded above by S,. Hence S must be unbounded below. Thus, given any MEIR, There exists a natural number N Such That Sy