I have that the application $$ d(f,g) := \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{||f - g||_{L^{\infty}(K_n)}}{(1 + ||f - g||_{L^{\infty}(K_n)})}, \forall f,g \in C(\Omega) $$ is a metric in $C(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is an open set and $\{K_n\}_{n \in \mathbb{N}}$ is a sequence of compact sets such that $\Omega = \bigcup_{n=1}^{\infty} \overset{\circ}{K_n}$. Given a Cauchy sequence $\{f_k\}_{k \in \mathbb{N}}$ in $(C(\Omega), d)$, I have to prove that this is also a Cauchy Sequence in $(C(J), ||\cdot||_{L^{\infty}(J)})$, where $J$ is a compact set contained in some $K_{n_0}$.
Any help is very welcome.
Let $(f_k)$ be a Cauchy sequence in $(C(\Omega),d)$ and $J$ a compact subset of $K_{n_0}$. Let $\varepsilon > 0$. Choose $\delta > 0$ such that $$ 2^{n_0}\delta < 1 ~~ \text{and} ~~ \frac{2^{n_0}\delta}{1-2^{n_0}\delta} < \varepsilon. $$ Then there exists $k_0$ such that $d(f_k,f_l) < \delta$ $(k,l \ge k_0)$. Hence, for $k,l \ge k_0$ we get
$$ \frac{\|f_k-f_l\|_{L^\infty(K_{n_0})}}{1+\|f_k-f_l\|_{L^\infty(K_{n_0})}} < 2^{n_0} \delta, $$ so $$ \|f_k-f_l\|_{L^\infty(J)}\le \|f_k-f_l\|_{L^\infty(K_{n_0})}<\frac{2^{n_0}\delta}{1-2^{n_0}\delta} < \varepsilon. $$