How to prove $P[X\geq u]\geq \frac{1}{2}(1-\sqrt{1-e^{-u^2}})$ when $X$ is standard normal?

Question

If $X$ is a standard normal random variable, I read that $$P[X\geq u]\geq \frac{1}{2}(1-\sqrt{1-e^{-u^2}}).$$

How to prove that?