Boundedness in real analysis

Question

I am struggling to come up with this book example question in my real analysis textbook:

Give an example of an ordered field in which {1,1+1,1+1+1,...} is bounded.

If the set of natural numbers is infinite, then I'm not sure if it is bounded.

Answer 1

The most accessible example is the field of rational functions with real (or rational) coefficients.

Let $f(x)=\dfrac{a_mx^m+a_{m-1}x^{m-1}+\cdots+a_0}{b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0}$ be such a function. We declare $f(x)$ to be positive if $\dfrac{a_m}{b_n}$ is positive.

One can verify that this is a field ordering. Then the rational function $x$ is greater $1$, $1+1$, $1+1+1$, and so on since $x-1$, $x-2$, $x-3$, and so on have positive lead coefficient. Thus the field element $x$ is an upper bound for $\{1,1+1,1+1+1,\dots\}$.

Remark: There are fancier examples. For instance, one can take the field $\mathbb{R}$. Let $D$ be a non-principal ultrafilter on the set $I$ of positive integers. Then the ultrapower $\mathbb{R}^I/D$ has a natural ordered field structure, and contains objects greater than $1, 1+1,1+1+1,\dots$. The advantage of this type of structure is that it is much more "$\mathbb{R}$-like" than the field of rational functions. For one thing, it is real-closed.