I am trying to understand why an inequality in the following post is true: Bounding rows of random matrices.
In particular, I am wondering why we have
$$ \exp\left(-\frac{t^2/2}{2np+t/3}\right)\le \exp\left(-\frac{Ct^2}{np}\right)+\exp(-Ct) $$
for some absolute constant $C$. Here $t > 0$, $n \in \mathbb{N}$, $p \in (0, 1]$. Perhaps this follows from the convexity of exp, but I am not seeing how after a couple attempts.
Its much simpler than convexity. $$ \exp\left( - \frac{t^2/2}{2np + t/3}\right)\le \exp\left( - \frac{t^2}{2\max(2np, t/3)}\right) = \exp\left(- \min\left( \frac{t^2}{4np} , \frac{3t}{2} \right) \right)\\ \le \exp\left( - \min\left( \frac{t^2}{4np}, \frac{t}{4}\right)\right)\le \exp( - t^2/4np) + \exp(-t/4),$$ so the claim holds with $C = 1/4$.