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All I have to do is show from definition that $A$ must have at most one greatest lower bound if $A$ is a subset of $\mathbb{R}$ and is not empty.

My thoughts are if $A$ is not bounded below, then it has no lower bound so no greatest lower bound by definition

And if $A$ is bounded below then there exists a unique $l$ which is a lower bound such that all lower bounds $m$ of $A$ are less than or equal to $l$.

This is just the definition so it doesn't seem right to me, is this Ok? Thanks

Boshu
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Peterpan23
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1 Answers1

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Suppose there are two greastest lower bound, $a$ and $b$.

Then for any lower bounds of $A$, $x$, we have $x \le a$. Since $b$ is a lower bound, we have $b \le a$.

By symmetry, we have $a \le b$.

Hence $a=b$.

Siong Thye Goh
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