Property of fractions Given two fractions $\frac{h}{k}$ and $\frac{h^{'}}{k^{'}}$ both in reduced form. I am unable to find a case when $\frac{h+h^{'}}{k+k^{'}}$ does not lie in the interval $\big[ \frac{h}{k},\frac{h^{'}}{k^{'}} \big]$. Is there such a case ?
PS: I was able two prove no such case exists for consecutive terms of Farey series. But can't prove in general.

Let's prove that, for positive $a,b,c,d$, with $\frac{a}{b}\le\frac{c}{d}$, it holds
$$
\frac{a}{b}\le\frac{a+c}{b+d}\le\frac{c}{d}
$$
The inequality on the left is equivalent to
$$
ab+ad\le ab+bc
$$
that is, $ad\le bc$, which is true.
The inequality on the right is equivalent to
$$
ad+cd\le bc+cd
$$
that is, $ad\le bc$, which is true.
Note that, if $a/b<c/d$, the two inferred inequalities are strict too. The hypothesis about the fractions being in reduced form is redundant.
