Could someone please check my work

Suppose ( Sn) is a convergent sequence with lim S, = S, Then Prove lim ( K. Sn) = KS. Ler Eso. Since ( Sn) converges, we know That ( S,) is bounded; That is, There exists some CERR Suck That ( SAJ EC for all n. 2+ 1 Ler &, =and E, = 2. Since E, 20 There exists some No such That JK-1/ N2 . Ler N= max EN., N. 3 . Then, for any n> N, ( K. Sn - KS ) = / K. Sn - Sn + Sn - KS) Sn- Sn/+ /Sn - ks/ by The mangle inequality K converges to do S, converges To KS one = [K-1//Sn) + Sn- ks/ K- 1 because