Indeed, here is a quick proof: let x be any real number that is supposed to be an upper bound for the set of integers. Writing x in its usual decimal notation, we see that x is of the form n+c, where n is some integer, and c, the fractional part of x, lies between 0 and 1 (and does not equal 1). Once we have done this, it is obvious that x is not an upper bound for the set of integers, since n+1 is bigger than x.
Is there anything wrong with the above argument? In one sense no, because the deduction as presented is valid. However, it rests on the assumption that every real number has a decimal expansion, and in most presentations of the theory of real numbers, this is considered less basic than the statement that the integers are unbounded. (I say most presentations, because it is possible to define the real numbers through their decimal expansions and then to prove that they form a complete ordered field. However, even then one would like to prove that there is only one complete ordered field, and for this the unboundedness of the integers, or some obviously equivalent statement, is important.)
One way to remove the feeling of obviousness from the statement is to reformulate it: in every complete ordered field F the integers (which could be defined as the additive subgroup of F generated by the multiplicative identity) form an unbounded set. This reformulation encourages us to think about the question abstractly, rather than relying on our normal, concrete representation of the real numbers.
Is it obvious in F how to associate with every element a decimal expansion? Here is how one might do it. Let x be an element of F, and for simplicity let us assume that x is positive. Let n be the largest integer (defined as above) that is smaller than x. For example, if 1 < x, 1+1 < x and 1+1+1 > x, then we will set n=2. Now let a1 be the largest integer m such that m/10 < x-n, then let a2 be the largest integer m such that m/100 < x-n-a1/10, and so on. The decimal expansion of x is then n.a1a2...
The above process works, in that all the concepts make sense in F (and not just in our familiar picture of R), but at the very first stage, when we pick out n, it relies on the fact that there is a largest integer n that is smaller than x. But what if x is an upper bound for the set of all integers? What would n be then?
You may wish to say, `But the integers are obviously unbounded, so this problem does not arise', but then you are appealing to a principle which itself seems obvious only once you know that every number has a decimal expansion. Indeed, the discussion so far shows that these are equivalent statements. How can we break out of the deadlock? Since decimal expansions are a little complicated, it is surely better to try to prove that the integers are unbounded.
Here is the usual proof: suppose that the integers were bounded in F. By the completeness axiom, there would have to be a least upper bound, x, say. Since no number smaller than x is an upper bound, there must be an integer n greater than x-1. But then n+1 is greater than x, contradicting the fact that x is an upper bound.
Was it really necessary to use the completeness axiom in the above proof? Yes it was, and the way to demonstrate this is to give an example of an ordered field in which the integers, still defined as the additive subgroup generated by the multiplicative identity, are bounded. Obviously the completeness axiom will have to fail for such a field.
The most familiar example of an ordered field failing the completeness axiom is Q, the field of rational numbers. However, the integers are certainly unbounded in Q. Indeed, if we want to make the integers bounded, it is clear that what we should be doing is to enlarge the reals somehow (to introduce an upper bound) rather than shrink them.
There are many ways to do this, of which here is one. Let F be the field of all rational functions in an indeterminate x - that is functions of the form p(x)/q(x), where p and q are polynomials (with real coefficients) and q is not identically zero. I do not insist that these functions should be everywhere defined - for now they can be thought of as formal expressions - though in fact they will be defined at all but finitely many points. Strictly speaking I am talking about equivalence classes of these formal expressions, so I regard p/q as the same as r/s if ps is the same polynomial as qr.
Such functions can be added and multiplied in the obvious ways, so it is not hard to check that F really is a field. We would now like to show that F is an ordered field, by defining a suitable ordering on the functions p/q. This we can do by saying that p/q is less than r/s if there exists a d > 0 such that p(x)/q(x) is less than r(x)/s(x) for every x in the open interval (0,d). Loosely speaking, we say that one rational function is smaller than another if it is smaller `near zero'.
Various facts about this ordering must be checked. One basic one is that this is a total ordering - that is, given two functions p/q and r/s, then either they are equal or one is less than the other. To see this, note that if neither is less than the other, then as x approaches zero from above, the sign of p(x)/q(x) - r(x)/s(x) must change infinitely often, so that p(x)/q(x) - r(x)/s(x) is zero infinitely often. But this means that the polynomial ps-qr has infinitely many roots, and is therefore the zero polynomial, which implies that p/q equals r/s. The other axioms for an ordered field are easy to check.
What are the `integers' in such a field? Well, the multiplicative identity is the constant function 1, and the subgroup it generates consists of all integer-valued constant functions. Thus, in this field, 26 means the function that always takes the value 26, and so on. What would be an upper bound for all of these functions in the ordering just defined? An obvious example is the function 1/x, which, if you are near enough to zero, is at least as big as any fixed integer.
Notice also that in this ordered field there are some very small elements as well. We can identify the real number t with the constant function f(x)=t, and then the function f(x)=x is smaller than all real numbers, but not zero. Thus, for example, the sequence 1/n does not converge to zero in this field (because if we let epsilon be the function f(x)=x, then epsilon > 0 and there is no constant function 1/n which is smaller than epsilon).
Two further remarks: (i) in the field F it is not true that every element has a decimal expansion; (ii) we could have defined the ordering by saying that one function is smaller than another if it is smaller for sufficiently large x instead, and the argument, with obvious changes, would have worked just as well.
This example shows that it is not wholly obvious that the integers are unbounded in R. Any proof must use the completeness axiom in some form, because otherwise it will not distinguish between R and the ordered field F just constructed in which the integers are bounded.