Uniqueness of least upper bounds Show that if M1 and M2 are...

Uniqueness of least upper bounds Show that if $M_{1}$ and $M_{2}$
are least upper bounds for the sequence $\left\{a_{n}\right\},$ then $M_{1}=M_{2}$
That is, a sequence cannot have two different least upper bounds.

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Key Concepts

Least Upper Bound

The least upper bound (or supremum) of a set is the smallest number that is greater than or equal to every element in the set. It is defined by two properties: first, it must be an upper bound of the set; second, any number less than it cannot serve as an upper bound. This concept is fundamental in real analysis and is essential in the study of limits and completeness.

Upper Bound

An upper bound of a set is a number that is greater than or equal to every element of the set. Recognizing what qualifies as an upper bound is key in determining the supremum of a set, as the least upper bound must be chosen from among all possible upper bounds.

Uniqueness Argument

The uniqueness of the least upper bound is established by the definition of being the smallest of all upper bounds. If there were two distinct least upper bounds, by the comparability property of real numbers, each would have to be less than or equal to the other, which forces them to be equal. This argument underlines the non-ambiguity of the supremum in the real number system.

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