If $x_1 = (x_0 + 2)/(x_0 + 1)$, can we show $\sqrt{2}$ lies between $x_0$ and $x_1$?

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…without using tables, show that each of \(\dfrac{41}{29}\) and \(\dfrac{99}{70}\) differ from \(\sqrt{2}\) by less than \(5 \times 10^{-4}\).

Without knowing \(\sqrt{2}\), we can’t calculate the distance between \(x_n\) and \(\sqrt{2}\), but what’s the distance between \(x_n\) and \(x_{n+1}\)?