Let $f,g: \mathbb{R} \supset [a,b] \rightarrow \mathbb{R}$ be continuous with g non-zero. Suppose $(f_n)_1^\infty$ and $(g_n)_1^\infty$ are sequences of functions defined on $[a,b]$ that converge uniformly to $f$ and $g$ respectively.
- Prove that for all sufficiently large $n$, $1/g_n$ is defined, and that $(f_n/g_n)$ converges uniformly to $f/g$ on $[a,b]$.
- Show that the above may not hold if we replace [a,b] with (a,b).
I am only confused on the first part of (1). I don't know why we would need to prove this and don't understand what exactly they are asking. Isn't $1/g_n$ defined because we are given that $(g_n)$ is non-zero?
The trick: because the continuity of $g$ and the compacness of $[a,b]$, $\exists k>0:~g(x)\ge k$ or $g(x)\le -k$. WLOG, suppose $g(x)\ge k$. And the $g_n$...