Assume I have two small parameters $0< \epsilon \ll 1$ and $0< \delta \ll 1$ such that $\delta \gg \epsilon^{2}$. I would like to find an upper bound for the quantity $A = O\left(\frac{\epsilon^{2}}{\delta^{2}}\right) \times O\left(\frac{1}{\epsilon}\right)$, and more precisely I am hoping to find that $A \ll \frac{C}{\epsilon^{2}}$ for some constant $C$.
Now since from above we have $A = O\left(\frac{\epsilon}{\delta^{2}}\right)$ and using $\frac{1}{\delta} \ll \frac{1}{\epsilon^{2}}$ twice, we can write
$A = O\left(\frac{\epsilon}{\delta^{2}}\right) \ll \frac{1}{\epsilon \delta} \ll \frac{1}{\epsilon^{3}}$
so I have bounded $A$ by $\epsilon^{-3}$ - but can I do better and show that if it is true that $A \ll \frac{C}{\epsilon^{2}}$ ?