Well-order

The standard ordering ≤ of the natural numbers is a well ordering and has the additional property that every non-zero natural number has a unique predecessor.

Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:

This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Two elements lack a predecessor: 0 and 1.

If a set is totally ordered, then the following are equivalent to each other:

A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum which is also maximum of the whole set.