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Step-by-step solution

Step 1 of 3

Given a real number a, deﬁne

. By the deﬁnition of S, a is an upper

bound of S. We will show that a is the least upper bound of S by assuming that this is false and
deriving a contradiction.

NEXT

Comment

Step 2 of 3

Let c be an upper bound of S such that
there is a rational number x in
Moreover, since

. Since the set of rational numbers is dense in

. Therefore, x is a rational number and

,

, so x is in S.

, c is not an upper bound of S as originally assumed.

Comment

Step 3 of 3
This contradiction implies that no number less than a is an upper bound of S. Thus,

Comment

.

Related keywords

solution number originally there dense comment upper rational bound since moreover therefore contradiction assumed numbers

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