If we let and let be a continuous function on such that for each in there exists in such that . Prove there exists a point in such that .
Additionally, you could suppose that on the interval (if you can't do this then f is the zero function and there is nothing to prove).
Then choose a point , so you have a new point such that .
The by induction you have points which satisfy
Now some subsequence converges to a point . And since is continuous
Really, the wasn't important. You just needed a constant and the same proof will work.
f has a root in [a,b] if and only if |f| has a root in [a,b], right?
If |f| doesn't have a root in [a,b], then 0 < |f(x)| for all x in [a,b], right? From that assumption, I derived a contradiction. Which must mean that there indeed does exist a point x = c such that 0 = |f(c)| = f(c)