If Xn->X (in probability) and Yn->Y(in probability) and Cn=Xn+Yn and C=X+Y show that Cn->C (in probability).

what i got so far

|Xn-X|<=ε/2 and |Yn-Y|<=ε/2

|(Xn+Yn)-(X+Y)|=|Xn-X+Yn-Y|<=|Xn-X|+|Yn-Y|<=ε/2+ε/2=ε

|(Xn+Yn)-(X+Y)|<=ε

|Cn-C|<=ε

so lim(n->inf) P(|Cn-C|>=ε)=0

I want to verify if this is correct, and if not, what's the mistake, or if I am missing anything.

Thanks