If $\mathcal C_{\mathcal C}(\mathbb R)$ is the set with continuous functions $f: \mathbb R \to \mathbb R $ so that there exists a $M \in \mathbb R^+$ (which depends on $f$) so $f(x) = 0$ for every $x \in \mathbb R$ with $|x|>M$. We see that the functions in $\mathcal C_{\mathcal C}(\mathbb R)$ are bounded so we can use the supremummetric $d_{\infty}$ on $\mathcal C_{\mathcal C}(\mathbb R)$, with $$d_{\infty}(f,g) = sup\left\{d(f(x),g(x))\mid x \in X\right\}.$$ But with this metric it isn't complete, but what is than the completion of $\mathcal C_{\mathcal C}(\mathbb R)$? How can we make this space with this metric complete?
Thanks in advance
The completion is the following space $$C_0(\mathbb{R}):=\left\{f\in C(\mathbb{R}):\lim_{|x|\to \infty}f(x)=0\right\} $$ Notice that $C_0(\mathbb{R})$ is closed in $C_b(\mathbb{R})=C(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, which is a Banach space, so $C_0(\mathbb{R})$ is a Banach space. (Check that if $\left\{f_n\right\}$ is such that $\lim_{|x|\to \infty}f_n(x)=0$ for all $n$, and $\sup_{\mathbb{R}}|f_n-f|\to 0$ for some $f\in C_b(\mathbb{R})$, then also $\lim_{|x|\to \infty}f(x)=0$).
Moreover, $C_c(\mathbb{R})$ is dense in $C_0(\mathbb{R})$ - for each $f$ in $C_0(\mathbb{R})$ you may construct an approximating sequence of $C_c(\mathbb{R})$ functions $\left\{f_n\right\}\to f$ obtained by truncating the function at $[-n,n]$ and smoothing at the endpoints in order to preserve continuity without affecting convergence (you may formalize this using convolution by mollifiers).