The Archimedean Principle:

Let x and y be positive real numbers. 

Then there is some positive integer n such that y < nx.

Proof:  Suppose that this is false; that is to say, suppose nx £ y for all natural numbers n.  In this case the set of numbers defined by nx is bounded above, that is to say that there is at least one number a, called an upper bound, for which nx £ a for all values of nx.  Let us consider the ‘least upper bound’, or supremum, which we will denote by s and define here as the upper bound which is less than or equal to all others. 

Now, it is clear to see that s - x < s , and that we must have some element of the set of numbers defined by nx in between, i.e.  s – x < nx £ s  for some n.  From this we see that s < (n + 1)x, but (n + 1) is a natural number (remember, our hypothesis for this proof is that nx £ y for all natural numbers), and so (n + 1)x is a member of the set defined by nx which has s as its least upper bound and so cannot be greater than s. 

Clearly, we have arrived at a contradiction and our hypothesis must be incorrect.  In other words, if it is not true that, for all values of n, where n is a natural number,

nx £ y, then it must be the case that nx £ y for some n.  This proves the Archimedean Principle.

Now, you may well wonder why we have gone to the bother of proving such a seemingly obvious point.  If you are wondering this, I am shocked, quite frankly, at both your ingratitude and your lack of wonder and curiosity.  We’re not in the world of physics any more, Mathematics isn’t a case of just pointing at something like the orbit of a planet, saying “yeah, well, from here it kind of looks like it’s following an elliptical path” and then swanning off to collect your medal from some scientific society or something!  It doesn’t matter how ‘sure’ you were previously that the Archimedean Principle holds, now you know that it holds and have another principle to mention whenever you feel the need for an ego boost.  Of course, it is also the case that, if you weren’t already familiar with it before, you have just been introduced to another method of proof.  The one demonstrated above is the beautifully named reductio ad absurdum (for barbarians, it is also known, less beautifully, as proof by contradiction), and is one of the more elegant forms of proof available to the mathematician.  I believe I feel another Hardy quote coming on…“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”. 

In short, if all the evidence available to you suggests that a hypothesis is true a useful method of proof is to:

i)                    Assume that the contrary hypothesis is true.

ii)                  Prove that this hypothesis leads to a contradiction, and is therefore false.

Also, it must be said that, for such a simple observation, The Archimedean Principle has a great many uses, a couple of which we are about to see.