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\begin{align*}
\frac{8m}{2+m}+\frac{9n}{1+n}&=8-\frac{16}{2+m}+9-\frac{9}{1+n}\\[1ex]
&=17-\left(\frac{64}{8+4m}+\frac{81}{9+9n}\right)\\[1ex]
&\leqslant17-\frac{(8+9)^2}{(8+4m)+(9+9n)}\\[1ex]
&=\frac{17}{18},
\end{align*}
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