The procedure starts by letting $m = \frac {a+b}2$.

If $f(m) f(a) < 0$ then the next [smaller] interval to consider is $[a,m]$.

If $f(m) f(a) \gt 0$ then the next [smaller] interval to consider is $[m,b]$.

Following this procedure, the hypothesis for the intermediate value theorem can be applied to a sequence of intervals where the midpoints for each interval will converge to a single number $m_*$.

By the continuity of $f$ on $[a,b]$, it can be seen that $f(m_*)$ cannot be positive or negative, so $f(m_*)=0$.

Thus there is a number, $c = m_*$, with $a < c < b$ where $f(c)=0$.

Below is a GeoGebra visualization of the bisection method proof of the Intermediate Value Theorem CCCD.IVT.0 using mapping diagrams applied to $f(x)= x^2-2, a =0,b=2.$

Notice how the points on the graph are paired with the points on the mapping diagram.

Martin Flashman, 3 June 2015, Created with GeoGebra