Let $C(\mathbb{R})$ be the set of continuous functions on $\mathbb{R}$. Let $w(t) = \frac{t}{1+t}$ for $t \geq 0$. Define $$d(f,g)= \sum^{\infty}_{k=0} 2^{-k} w(\text{sup}_{[-k,k]} |f(x)-g(x)|).$$
Show that $C(\mathbb{R})$ is complete with this metric.
I believe it comes down to constructing a limit function which converges in $\mathcal{C}(\mathbb{R})$ under the supremum norm, but since the function need not be bounded over all of $\mathbb{R}$, I am having trouble constructing such a function
If $\{f_n\}$ is Cauchy then $w(\sup_{[-k,k]} |f_n(x)-f_m(x)|) \to 0$ for each $k$. It is easy to see from the definition of $w$ that $\sup_{[-k,k]} |f_n(x)-f_m(x)|) \to 0$ for each $k$. Hence $f(x) =\lim_{n \to \infty} f_n(x)$ exists for each $x \in [-k,k]$ and since $k$ is arbitrary, $f(x)$ is defined for all real numbers $x$. For proving that $d(f_n,f) \to 0$ here is a hint: given $\epsilon >0$ there exists $N$ such that the sum from $k=N+1$ to $\infty$ in the definition of $d(f_n,f)$ is less than $\epsilon$ (because $w \leq 1$. $f$ is continuous because on each of the intervals $[-k,k]$ it is a uniform limit of continuous functions.