# Show that $(C,d)$ is a completion of $(\Phi,d)$

Let $\Phi= \{ a_n \in R : a_n=0$ after some $n \}$ be equipped with the sup metric $d$ defined by $d( a_n, b_n) = \sup_n \lvert a_n-b_n \rvert$.

We are supposed to prove that $C := \{x_n \in R : \lim x_n =0 \}$ with the same metric $d$ is a completion of $(\Phi, d)$. [Note that you do not need to prove $(C,d)$ is complete.]