Suppose I am given some set/interval, e.g. $A = [5,6[$.
I need to find $minA$, $maxA$, $infA$, $supA$, which is pretty easy in this case. $minA=infA=5$, $supA=6$, and $max$ doesn't exist. But how can I prove my assumptions. I've got some kind of idea, like e.g. for $inf$ asuming there is a bigger lower bound, and then showing that this isn't possible. But I am not really sure in my proving skills, I'd really appreciate if someone could show me the way to prove all those assumptions.
Thanks in avance
EDIT: here is one of my approaches, I am grateful for every correction, no matter how small it may be:
Proving $infA$
Suppose there exists a greater lower bound than 5, in this case:
$(x\in\mathbb{R}:x>5\,\,\land \forall_{a\in A}x\leq a) \Rightarrow \exists_{k\in\mathbb{R}}:x=5+k$
Here is my problem, in solving such trivial matters. I know I need to show that $x$ lays in $A$, and that it is greater, than 5, so it can't be a lower bound, but I feel like missing something?
Assume $5+\epsilon$ is lower bound for $\epsilon>0$ therefore $$5+\dfrac{\epsilon}{2}<5+\epsilon$$which is in contrary with our primary assumption.