Let $C[0,\infty]$ be the space of all continuous functions on $[0, \infty ]$ with metric $$ \phi(\omega_1, \omega_2) = \sum^{\infty}_{n=1} \frac{1}{2^n}\max_{0{\leq} t {\leq} n}(|\omega_1(t)-\omega_2(t)|\wedge 1)$$ where $n \in \mathbb{N}$. Show that the space $C[0,\infty]$ under $\phi$ is complete and separable.
I proceeded as follows: suppose I have a Cauchy sequence $\{\omega_n\}$ in $C[0,\infty]$. Its limit exists in $\phi$ as $\phi$ is bounded. Let the limit be $\omega$, now I have to show that $\omega $ belongs to $C[0,\infty]$.
Notice that for each $N$, $\sup_{0\leqslant x\leqslant N}|\omega_n(x)-\omega(x)|\to 0$ as $n$ goes to infinity and a uniform limit of continuous functions is continuous. This proves that $\omega$ is continuous on $[0,N]$ for each $N$, hence $\omega$ is continuous on $[0,\infty)$.
To conclude the proof of completeness, it remains to check that $\phi(\omega_n,\omega)\to 0$.
For separability, notice that $C[0,N]$ is separable for each $N$.