I am reading a proof on subguassian tails and I am confused by the estimation I just read, can anyone explain how it makes sense?
Let F be the distribution function of the random variable X. What the proof is calculating is $ \mathbb{E} [e^{uX}] = \int_{- \infty}^{\infty} e^{ut} dF(t) $. First, the proof separates this calculation from $- \infty$ to $\frac{1}{u}$ and from $\frac{1}{u}$ to $\infty$. I have problems with the latter part.
$$ \int_{\frac{1}{u}}^{\infty} e^{ut} dF(t) \leq \sum_{k=1}^{\infty} e^{k+1} \operatorname{Prob}\left[X \geq \frac{k}{u} \right] \leq \sum_{k=1}^{\infty} e^{2k}e^{-ak^2/u^2} = \sum_{k=1}^{\infty} e^{k(2-ak/u^2)}$$.
I don't understand the less than approximation happening at $ \int_{\frac{1}{u}}^{\infty} e^{ut} dF(t) \leq \sum_{k=1}^{\infty} e^{k+1} \operatorname{Prob}[X \geq \frac{k}{u}] $. How does that make sense?
This treats the integral as a sum of the integrals over the intervals $\left[\frac ku,\frac{k+1}u\right]$, bounds the integrand in each interval by its value at the upper endpoint of the interval and bounds the measure of each interval by the entire remaining measure up to $\infty$.