Could someone please check my work

1. Suppose that lim S = 0 . If (tn) is a bounded sequence, prove that lim (Sn . tn) = 0. Suppose lim Sn = 0 and (tn) is a bounded sequence. Since (tn) is bounded, 3m > 0 such that |tn) 0, 3N E N such that Vn > N, Isn - 0| lal . 125/ 5 12071 . 1251 = 127. 281 = 10- 27. us/'N 0, N E N such that Vn > N , Isn - 0| = |snl N, Isn . tn - 0| = Isn . tal = Isnl . Itn)