Prove that if S proper subset 0 infinity is a wellordered se

Prove that if S proper subset [0, infinity) is a well-ordered set, then T={x/(x+1):x belongsto S} is a well-ordered subset of [0,1].

Solution

since x is from S,so x >=0

if x=0 then x/(x+1) = 0

if x>0 then x<x+1 ,so x/(x+1) <1

we saw that if x is from S then x/(x+1) is either 0 or less then 1.so x/(x+1) is from [0,1).

since any element of T looks like x/(x+1) so obviously T is a subset of [0,1).