To show $f$ has a root using **Bisection Method**

Question

let $f:[a,b]\rightarrow \mathbb{R}$ be a continous funtion such that $f(a)f(b)<0$.

Question: Show that, using Bisection Method, $f$ has a root.

I know using the Continuity of f and Intermediate Value Theorem we can show that $f$has a root and using bisection method it's guaranteed to converge at that root.

But how to use the bisection method to show that $f$ has a root?

Answer 1

Using the bisection method, if it does not terminate early at a root of $f$, you get a growing infinite sequence $a_n$ and a falling sequence $b_n$ with $a_n<b_n$, $\lim a_n=\lim b_n=x_*$ and $f(a_n)f(b_n)<0$. Now by continuity $f(x_*)^2\le 0$ which is only possible if $x_*$ is a root.