Let $f,g$ be continuous on a closed bounded interval $[a,b]$ with $|g(x)| > 0$ for all $x$ in $[a,b]$. Suppose that $f_n \to f$ and $g_n \to g$ uniformly on $[a,b]$. Prove that $1/g_n$ is defined for large $n$ and $f_n/g_n \to f/g$ uniformly on $[a,b]$.
I have already shown that $1/g_n$ is defined for a large $n$. Note that we can conclude that either $g(x) > 0$ or $g(x) < 0$ for all $x \in [a, b]$, which follows by the Intermediate Value Theorem. Suppose $g(x) > 0$. Let $|g(x)| > \varepsilon_0 > k > 0$ for all $x\in [a,b]$. Since $g_n \to g$ uniformly, there exists $N \in \mathbb{N}$ such that if $n \ge N$ and $x \in [a,b]$, we have $|g_n - g| < \varepsilon_0$. This implies that $0 < k - \varepsilon_0 < g(x) - \varepsilon_0 < g_n$. Thus, we show $1/g_n$ is defined for a large $n$.
I struggle to show $f_n/g_n \to f/g$ uniformly. Any help will be appreciated. I know someone posted the same question but the answer is not complete. I know I need to use $g > k$ and $g_n > k - \varepsilon_0 \, \forall n \ge N$. But I am still struggle to show $|f_n/g_n - f/g| < \epsilon$.