Using the estimation lemma on semicircle

Question

I need to prove, using the estimation lemma, that

$\left| \int_{C_R} \frac{1}{z^2 + 2} dz \right| < \frac{R\pi}{R^2 - 2}$

Where $C_R$ is the semicircular curve in the upper half-plane from $R$ to $−R$.

I know that $l(C_R) = \pi R$

Unsure of how to evaluate $\max |f(z)|$ along the semicircle.

Answer 1

$|z^2 + 2| \geq |z^2| - 2 = R^2 - 2$ for $|z| = R$.