Real cosine function/One zero between 0 and 2/Fact/Proof

Proof

We consider the cosine series

For , we have . For , one can write

Hence, due to the intermediate value theorem, there exists at least one zero in the given interval.
To prove uniqueness, we consider the derivative of cosine, which is

due to fact. Hence, it is enough to show that sine is positive in the interval , because then cosine is strictly decreasing by fact in the interval and there is only one zero. Now, for , we have