Some Babylonian trick

Question

My teacher used some trick for proving there is no supremum in $\mathbb{Q}$ for $\{x|0<x^2<2, x \in \mathbb{Q}\}$.

He defines a variable: $\xi=\frac{2c+2}{c+2}$. So, I noticed that $\xi=\sqrt{2}$ if $c=\sqrt{2}$. Where does this $\xi$ comes from?

Answer 1

The expression for $\xi$ as $\frac{2c+2}{c+2}$ is just a convenient rational function of $c$ which is larger than $c$ when $0<c^2<2$, and also satisfies the same inequality - i.e $0< \xi^2 < 2$.

This can then be used to get a strictly increasing sequence of values in the set $\{x \mid 0<x^2<2\}$ which tends to the limit $\sqrt{2}$ without ever reaching $\sqrt{2}$.

There are many other possible expressions for $\xi$ as a function of $c$ which would work as well, so as to where this particular function comes from, I guess it might have been found by some Babylonian (although I didn't think they had algebra developed sufficiently, but I might be wrong!)

Edit:

Another example with similar properties that you might like to look at could be $\xi = \frac{3c+4}{2c+3}$.