Explanation of why a certain integral-inequality is true

Question

While reading about Laplace Transform" (Fourier Analysis and Its Applications, Anders Vretblad), I came across the following theorem and proof:

What is behind this inequality and why is it true?

Answer 1

Since $s > s_0$, $t \mapsto e^{-(s - s_0)t}$ is a decreasing function in $t$. Thus the largest value for $t \in [\delta,\infty)$ is at $t = \delta$, so the integral inequality follows from the inequality $$e^{-(s-s_0)t} \leq e^{-(s-s_0)\delta}$$

for $t \in [\delta,\infty)$.