All countable chains have an upper bound or maximal elements?

Question

I have a question that,

All countable chains have an upper bound or maximal elements? IF a chain is finite, this statement is true.

But, I'm not sure in the case of when a chain is countably infinite.

So, I would refine my question

All countably infinite chains have an upper bound or maximal elements?

or

Does the set of natural numbers has an upper bound or maximal elements in range of natural numbers?

Answer 1

Clearly the natural numbers have no maximal element. Suppose that there is a maximal element $n$, then $n+1 > n$ a contradiction.