Prove that $\lim _{t \rightarrow \infty}\|f\|=0$

I am reading Paper but I have a difficult in understand the following statement

They proved that $\frac{1}{2} \frac{d}{d t}\|f\|^2=\frac{1}{2} f^{\prime}\left[G+G^{\prime}\right] f+\Sigma \bar{u}_j f^{\prime} Q_j f \leqslant-\varepsilon\|f\|^2$ for some $\epsilon >0$.

Then, they claim that from that equation, we can obtain $\displaystyle \lim _{t \rightarrow \infty}\|f\|=0$.

I don't know how to obtain $\displaystyle \lim _{t \rightarrow \infty}\|f\|=0$.

My attempt:

Case 1: If $\|f\|=0$, then obviously $\displaystyle \lim _{t \rightarrow \infty}\|f\|=0$.

Case 2: If $\|f\| \ne 0$, then $\frac{1}{2} \dfrac{\frac{d}{dt}\|f\|^2}{\|f\|^2} \le -\epsilon$. Then $\frac{1}{2} \ln \|f\|^2 \le -\epsilon t$ where $t >0$ (I am not pretty sure about that, I took integral two sides). As a result, then $\|f\|^2 \le e^{-2\epsilon t}$. Then $\displaystyle \lim _{t \rightarrow \infty}\|f\|^2=0$.

Could you please help me with checking my work?

p.s: For $Q$ and $u$. I think you don't need to think about it. You just need to use the first and the last equality.The mapping $f(x,u,r)$ of $E^{m+k} \to E^{m}$.

p.ss: "For some $\epsilon$" means $\exists \epsilon$.