Wait before you dismiss this as a crank question :)
A friend of mine teaches school kids, and the book she uses states something to the following effect:
If you divide the circumference of any circle by its diameter, you get the same number, and this number is an irrational number which starts off as 3.14159... .
One of the smarter kids in class now has the following doubt:
Why is this number equal to 3.14159....? Why is it not some other irrational number?
My friend is in a fix as to how to answer this in a sensible manner. Could you help us with this?
I have the following idea about how to answer this: Show that the ratio must be greater than $3$. Now show that it must be less than $3.5$. Then show that it must be greater than $3.1$. And so on ... .
The trouble with this is that
I don't know of any easy way of doing this, which would also be accessible to a school student. Could you point me to some approximation argument of this form?