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 Question from Aishwarya, a student: Prove that if the lub and glb of a set exist then they are unique

Hi Aishwarya,

Let S be a nonempty set of real numbers that has an upper bound. Then a number c is called the least upper bound for S iff it satisfies the following properties:

1. c>=x for all x in S.

2. For all real numbers k, if k is an upper bound for S, then k>=c.

Suppose $a$ and $b$ are least upper bounds (lub) of a set $S$ in the real line. By 1. above both $a$ and $b$ are upper bounds of $S.$ Since $a$ is a lub of $S$ what does 2. tell you about the relationship between $a$ and $b?$ Since $b$ is a lub of $S$ what does 2. tell you about the relationship between $b$ and $a?$

Penny

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