DES - Demonstration of the First Property of the Transition Rate Matrix Q

From the first part of the Demonstration on How to calculate the Matrix Q we know that:

$$H(t)=e^{Qt}=\sum _{n=0}^{\infty }\frac{(Qt{)}^n}{n!}$$

then for $dt\to 0$

$$H(dt)=I+Qdt+o(dt)$$

Where $o(dt)$ is defined as a quantity that goes to $0$ faster than $dt$

Then, for $i\ne j$:

$$p_{i,j}(dt)=0+q_{i,j}dt+o(dt)$$

but the $p_{i,j}(dt)$ is $\ge 0$ as we can see here $p$ is describing a probability, and a probability cannot be negative, then $q_{i,j}dt>0$.