so here is what i have attempted, suppose we have two rational numbers $s$ and $t$, $s = \frac{p}{q}$ and $t = \frac{m}{n}$
adding these together results in a number of the form $\frac{pn + qm}{qn}$, and this satisfies the definition of the rational number.
we can further show that integers are closed in multiplication, and addition, so the both denominator and numerator are integers as well.
I think that's enough, or do i need something else.