I'm reading Wainwright's High Dimensional Statistics and am currently trying to wrap my head around the proof of Proposition 2.14. At some point, the author states:
$$ e^{-\lambda \mathbb{E}[X]} \cdot \left(1 + \lambda \mathbb{E}[X] + \frac{1}{2} \lambda^2 \mathbb{E}[X^2] h(\lambda b) \right) \leq \exp\left(\frac{\lambda^2 \mathbb{E}[X^2]}{2} h(\lambda b)\right) $$ where $X$ is a random variable almost surely bounded above by $b$ and $h(u) := 2 \frac{e^u -u - 1}{u^2}$, and $\lambda > 0$. Where is this inequality coming from? It's easy to verify it is true when $\mathbb{E}[X] \geq 0$, but there is no such assumption in the proposition.
Full context:
