CCD.IVT.0 Proof: The Bisection Method can be used to prove CCD.IVT.0 by approximating $c$ using a midpoint $m$ repeatedly for intervals of decreasing length, $[a,b]$:
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 $f(x)= x^2-2, a =0,b=2.$

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

Martin Flashman, 3 June 2015, Created with GeoGebra

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