Assume $\epsilon$ is a small positive parameter, $0 < \epsilon <<1$, $k$ is another parameter (the size of which we don't know), and $B_{1}$ and $B_{2}>1$ are constants. Suppose I have a quantity $\Delta t$ satisfying the inequalities $B_{2} \epsilon \leq \Delta t \leq B_{1}k$.
Now assume I have some other quantity, call it $Q$, such that $Q \sim O((\Delta t)^{2} + \epsilon \Delta t)$. Do the above inequalities imply that I can write
$$O((\Delta t)^{2} + \epsilon \Delta t) \sim O(k^{2})$$
?
$$(\Delta t)^2\leq B_1^2k^2$$ $$\epsilon\leq\frac{B_1}{B_2}k$$ $$\epsilon\Delta t\leq \frac{ B_1^2}{B_2}k^2$$ Sum the first and the third inequalities to finish the proof.