What is Convergent sequence?

Example 20.22

Let $u_0 = 2$ and $u_{n+1} = \frac{1}{2}\left(u_n + \frac{2}{u_n}\right)$. One checks by induction that $u_n \geq \sqrt{2}$ for all $n$ (the inequality $\frac{1}{2}(x + 2/x) \geq \sqrt{2}$ for $x>0$ is equivalent to $(x - \sqrt2)^2 \geq 0$), then that $(u_n)$ is decreasing, since

Decreasing and bounded below, $(u_n)$ converges to some $\ell \geq \sqrt{2}$, which must satisfy $\ell = \frac{1}{2}(\ell + 2/\ell)$, i.e. $\ell^2 = 2$. Hence $u_n \to \sqrt{2}$. This is Heron’s algorithm, already used by the Babylonians; its convergence is extremely fast ($u_3$ already gives $\sqrt 2$ to eight decimal places).