<s>[INST] Given a positive real number $\varepsilon > 0$, let $f_\varepsilon(x) = \varepsilon x(1-x)$. Suppose we have two sequences of real numbers $\{x_n\}$ and $\{y_n\}$, such that $x_n \to 1$ and $y_n \to 1$ as $n \to \infty$, but $x_n \ne 1$ and $y_n \ne 1$ for any $n$. Provide a proof that $$\lim_{n\to\infty} \frac{f_\varepsilon(x_n)}{f_\varepsilon(y_n)} = 1$$ for all $\varepsilon > 0$.[/INST]