show that the set 0 to infinitethe nonnegative real numbers

show that the set 0 to infinite(the non-negative real numbers) does not satisfy the well-ordering principle. In other words,show that it contains a subset A which does not contain the smallest element.

Solution

The well-ordering principle states that every non-empty set of positive integers contains a least element.In other words, the set of positive integers is well-ordered.

The set of integers {…, 2, 1, 0, 1, 2, 3, …} contains a well-orderedsubset, called the natural numbers, in which every nonempty subset contains a least element.

In a well-ordered set S, every subset A which has an upper bound has a least upper bound, namely the least element of the subset of all upper bounds of A in S.

The standard ordering of any real interval is not a well-ordering, since, for example, the open interval (0, 1) [0,1] does not contain a least element.

The set {1/n : n =1,2,3,...} has no least element and is therefore not a well-order (again, under standard ordering ).

So as per the two examples stated above we can say that it doesnot contain a well order.