Maybe it's a simple question... Fix a positive $N$. Let $a_{n}$ be the recursive sequence:

$$a_{1} = N$$ $$a_{n}=a_{n-1}-(a_{n-1})^{\frac{2}{3}}.$$

How many steps do I have to complete in order to have $a_{n}\leq 1$? And, more general, changing $\frac{2}{3}$ by $\frac{2+\varepsilon}{3}$...

Do you have any hints? Thanks a lot