Suppose lim(xn)= a and lim(yn)= b.
Let Un= max(xn,yn) and Vn= min(xn,yn) for all n.
Prove that limUn and limVn exist and find these limits.
Having lots of trouble setting this up. Thank you for any help.
There are really two cases to consider: a=b and a<>b.
If say a<b then let e=(b-a)/4.
Then there is a positive integer K such that if n>=K then x_n is in (a-e,a+e) and y_n is in (b-e,b+e). In these cases it is easy to see that U_n=y_n and V_n=x_n.
What happens if a=b?