pi is greater than 3

Statement

$$\pi > 3.$$

Remarks

The cosine series alternates. At $x = 3/2$, its terms decrease in size from the term $x^4/4!$ on. So $\cos(3/2) \ge 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}$ at $x = 3/2$. This is about $0.070 > 0$. Since $\cos 2 < 0$, the zero of $\cos$ in $[0, 2]$ lies in $(3/2, 2)$. So $\pi/2 > 3/2$. Archimedes proved the sharper bounds $\frac{223}{71} < \pi < \frac{22}{7}$. Mathlib states the claim as Real.pi_gt_three.

Explanations

L1 human

We show that $\cos x$ is still positive at $x = 3/2$. So its first zero $\pi/2$ comes later, and $\pi > 3$.