Let A be a set, totally ordered under relation ≤. Let x ∈ A. Then x is a maximal element of A if and only if x is the greatest element of A.

Question

We're proving both implications, so this is my first implication

Proof: Let A be a set, totally ordered under relation ≤. Let x ∈ A. Then x is a maximal element of A if and only if x is the greatest element of A.---

I am really stuck. I don't know where to go from here. please help! thank you.

Answer 1

What does it mean to be maximal in this case? $m$ is maximal if for every other element in $a$ we have $a \leq m$. Now what does it mean to be the greatest element. $g$ is greatest if $a \leq g$ for any other $a \in A$. So, how do you proceed?