Question

Let X be a non-empty set of real numbers. If it has a greatest lower bound, then it must be unique

Answer 1

The Trichotomy Principle for real numbers states that for $x,y\in\mathbb R$ either $x<y$ or $x=y$ or $x>y$.

Let $X$ be a non-empty set of real numbers. Suppose there exists two such greatest lower bounds $a,b\in\mathbb R$  such that $a\le x$  and $b\le x$ for all $x\in X$. So $a$ and $b$ are both lower bounds for the set $X$ and in particular, since $a$  and $b$  are both greatest lower bounds, $a\le b$  and  $b\le a$  by the definition of a greatest lower bound. Hence by the Trichotomy Principle, we conclude $a=b$. Therefore, a greatest lower bound must be unique.