If $A \subset R$ and there is a lower bound for $A$, then there is greatest lower bound for $A$

Question

Hey I need to prove this Statement using definitions.

If $A \subset R$ and there is a lower bound for $A$, then there is greatest lower bound for $A$

My try:

I drew a number line a take the sets as defined above. And then I mark some lower bounds on the line. And it is clear that there is greatest lower bound for the subset.

How to prove it analytically Or mathematically?

Answer 1

I think you have to use the following axiom:

$(AX)$ If $M \subset \mathbb R$ and there is an upper bound for $M$, then there is a smallest upper bound for $M$.

For your proof define $M:= \{-a: a \in A\}$. Then $M$ has the properties in $(AX).$

Can you proceed ?