The convergence in probability does not say , what you have is for each epsilon>0, as n increases.
The way you approach this problem is in terms of sets. What can you say about in terms of ?
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
{w:|(Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε}
{w:|Cn(w)-C(w)|>=ε}
so
{w:|(Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε}
contains
{w:|Xn(w)+X(w)|>=ε/2}U{w:|Yn(w)+Y(w)|>=ε/2}
so
P(|Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε) >= P(|Xn(w)+X(w)|>=ε/2)
P(|Xn(w)+Yn(w))-(X(w)+Y(w))|>=ε) >= P(|Yn(w)+Y(w)|>=ε/2)
P(|Cn-C|>=ε) >= P(|Xn(w)+X(w)|>=ε/2)
P(|Cn-C|>=ε) >= P(|Yn(w)+Y(w)|>=ε/2)
Am I on the right track here?