Mathematical induction

Question

I've already proven the first part but in (ii) I managed to get until

$u_{n+1} = \sqrt{9-\epsilon} - 1$

I don't know how to show the approximation answer.

Answer 1

$u_{n+1} = 3\sqrt{1-\dfrac{\epsilon}{9}} - 1 = 3\left(1-\dfrac{\epsilon}{9}\right)^{\frac{1}{2}} - 1\approx 3\left(1-\dfrac{1}{2}\cdot \dfrac{\epsilon}{9}\right)-1= 2-\dfrac{\epsilon}{6}$