Well Ordering Principle confusion

Question

I am very confused about how the well ordering principle works and was wondering if anyone could help me out. So if we have a non-empty set of integers $A$ and if there exists $b ∈ A$ such that for all $a ∈ A$, $b ≤ a$, then $b$ is the smallest element of $A$. The thing that confuses me is what $a$ is. Is $a$ any arbitrary integer greater than or equal to $b$ or is $a$ the integer that can come right after $b$, i.e. $b+1$? Also, how would we go about proving that $A$ even has a smallest element to begin with, given that $b\in\ \mathbb{Z}$?

Answer 1

You said it yourself, $a$ is any member of $A$. There is a big difference between the integers and the naturals in this respect. The integers are not well ordered, so you can find subsets of the integers that do not have a smallest element. One example would be all the even integers. The naturals (whether you think they include $0$ or not) are well ordered, so any non-empty subset of the naturals has a smallest element.