I have a problem with the last step on the proof: wiki
\begin{equation} \int_a^b\left[\frac{\partial L}{\partial f} - \frac{d}{dx}\frac{\partial L}{\partial f'}\right]\eta(x) dx =0 \end{equation} where $ε= 0$
Then they apply the Fundamental Lemma of Calculus of Variations.
In order to use this lemma, $\left[\frac{\partial L}{\partial f} - \frac{d}{dx}\frac{\partial L}{\partial f'}\right]$ can not somehow contain $\eta(x)$ implicitly\explicitly.
My question
How do we know that $\left[\frac{\partial L}{\partial f} - \frac{d}{dx}\frac{\partial L}{\partial f'}\right]$ does not have $\eta(x)$? is it because we set $ε= 0$ and that all $\eta(x)$ vanishes?