I'm trying to solve the following recurrence relation
$$T(n)\le T\left(\frac{9}{10}n\right)+T\left(\frac{1}{5}n\right)+\text{O}(n)$$
According to book it should be that $T(n)=\text{O}(n^2)$. I tried to prove by induction. Assume that for any $k<n$ the claim holds, i.e $T(k)\le ck^2$. Now I have \begin{align*}T(n)\le T\left(\frac{9}{10}n\right)+T\left(\frac{1}{5}n\right)+\text{O}(n) \le \frac{81}{100}c n^2+\frac{1}{25}c n^2+\text{O}(n)\\=\frac{85}{100}cn^2+\text{O}(n)\le cn^2 \end{align*}
Is this correct?