Finding an upperbound confusion

Question

let $$f(x) = \dfrac{x^{p-1}(x-1)^p...(x-m)^p}{(p-1)!}$$

let $0 \leq x \leq m$, then :

$$|f(x)| \leq \dfrac{m^{mp+p-1}}{(p-1)!}$$

Hence

$$ \left | \sum_{j=0}^m a_je^j \int_0^j e^{-x} f(x) \ dx \right | \leq \sum_{j=0}^m |a_je^j| \int_0^j \dfrac{m^{mp+p-1}}{(p-1)!} \ dx = \sum_{j=0}^m |a_j e^j| j \dfrac{m^{mp+p-1}}{(p-1)!} \to 0 $$ as $ p \to \infty$

Now I don't understand how they got the first inequailty: where did the $e^{-x}$ go? ($a_j \in \mathbb{Z}$)

Answer 1

The used that $e^{-x} \leq 1$ for $x \geq 0$ and removed it (or alternately also used that if $ a < b$ and $c > 0$ then $ac < bc$. They also replaced $f$ by its upper bound.