Define d on R Times R by dx y inf1 x y Show that d is a met

Define d on R Times R by d(x, y) = inf(1, |x -y|). Show that d is a metric on R.

The infimum (inf) function of a subset S is partially ordered set is equal to the greatest element that satisfies the lower bound.

For d to be metric

1) d(x,y)>=0 and equal to 0 if x=y

d(x,y) = inf(1,|x-y|), hence it will be equal to either 1 or the value of |x-y|

d(x,x) = inf(1,|x-x|) hence it will be equal to 0, since the minimum is 0

hence it satisfies the first property of being a metric on R

2) d(x,y) = inf(1,|x-y|) which will be equal to 1 or |x-y|

d(y,x) = inf(1,|y-x|) which will be equal to 1 or |y-x|

hence it will also satisfy the metric property

since d(x,y) = d(y,x)

3) d(x,y) <= d(x,z) + d(z,y)

|x-y| <= |x-z| + |z-y|

Hence the function will also satisfy this property

Hence it is proved that d is a metric on R^2