I'm getting stuck with this problem:
Let $\mathbb{X}$ be a compact metric space and $f_n, g_n: \mathbb{X} \rightarrow \mathbb{R}$ be functions converge uniformly to $f, g: \mathbb{X} \rightarrow \mathbb{R}$, respectively. Show that $\dfrac{f_n(x)}{1+\vert g_n(x) \vert}$ converges uniformly.
So far, I have the following:
\begin{eqnarray*} \bigg\vert \dfrac{f_n(x)}{1+\vert g_n(x) \vert} - \dfrac{f(x)}{1+\vert g(x) \vert} \bigg\vert &=& \bigg\vert \dfrac{f_n(x)(1+\vert g(x) \vert) - f(x) (1+\vert g_(x) \vert)}{(1+\vert g_(x) \vert)(1+\vert g(x)\vert)} \bigg\vert\\ &\leq& \big\vert f_n(x) - f(x) + f_n(x)\vert g(x) \vert - f(x) \vert g_n(x) \vert \big\vert\\ &\leq& \big\vert f_n(x) - f(x)\big\vert + \vert g(x) \vert \big\vert f_n(x) - f(x) \big\vert + \vert f(x) \vert \big\vert g(x) - g_n(x) \big\vert \end{eqnarray*}
But I'm stuck from here. My main question is, can I say that $\vert g(x) \vert$ and $\vert f(x) \vert$ are bounded since $f_n$ and $g_n$ converge uniformly to them on a compact set? If these were continuous functions I feel like I would be okay in showing that they are bounded, since they would then be uniformly continuous functions on on a compact set. But I don't know how to deal with them since I'm not guaranteed they are continuous. Is this the correct approach?