Answer to Question #154873 in Calculus for cs dev

Question

Exercise 4. Let X be a non-empty set of real numbers. If it has a supremum, then it must be unique.

Answer 1

Let us assume that the non empty set $X$ has two supremum(if it exists).

Also let $u$ and $u'$ be two supremum.

Then two cases arises

Case 1: if $u$ is supremum of $X$ , then $u$ is an upper bound of $X.$

Then every number belongs to $X$ is less than or equal to $u.$

$\therefore u'\leq u$ [ as $u'\in X$ ]

Case 2: if $u'$ is supremum of $X$ , then $u'$ is an upper bound of $X.$

Then every number belongs to $X$ is less than or equal to $u'.$

$\therefore u\leq u'$ [ as $u\in X$]

Now these two cases hold only when $u=u'$ .

Therefore the supremum of a non empty set $X$ is unique.