Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / U / 1 / a / i / Solution

The bisection method begins with a finite interval $[a,b]$ on which the continuous function has opposite signs at its endpoints. The intermediate value theorem guarantees a root inside. Evaluate at the midpoint $m=(a+b)/2$. If $f(m)=0$, stop. Otherwise retain the half whose endpoints have opposite signs and repeat. An endpoint that is already a root is returned immediately. The nested intervals retain a root and their lengths halve, so their midpoints converge to a root.