A set is said to be a dense subset of if , , such that . My this note is to show that the set is dense in . For that we observe that satisfies the following properties:
i)Additivity: If , then .
ii)Homogeneity in : If and , then
Now we observe that , .
Also we note that ,
Now, let Then and . Also if , then .Hence is an infinite subset of .
Now we shall show that for any , such that . For that, given any , we partition into equal subintervals, each having size less than . Then, since is an infinite subset of , so at least one subinterval must contain two distinct elements of , say . Without loss of generality, let . Then obviously . Now, from the aforesaid properties i) and ii) of , we have that and hence . This shows that for any , such that .
Now let be a real to be chosen such that be given. Then it is trivial to show that such that . Now we take . Then such that . Hence, such that . Since by property ii), ,the property is proved.
There is another way to prove the above proposition. First prove that every proper subgroup of the additive group is either dense or cyclic in . Then prove that the given subset of is a subgroup of that is not cyclic in ..hence it is dense in .