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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?