I have a statement involvinga function $f(\tau)=\mathcal O(H(\tau))$, and I could show that this function $H(\tau)$ satisfies $\lim_{\tau\to\infty}H(\tau)=0$. So I can say that $H(\tau)=\mathcal o(1)$, correct?
If so, would this then imply that $f(\tau)=o(1)$ as well?
So phrased differently: Is $\mathcal O(o(1))=o(1)?$