For each positive integer $m$, consider the following sequences: $x_m:=\{0,0,...0,m,m,m,...\}$ and $y_m:=\{0,...,0,\frac{1}{m}, \frac{1}{m},...\}$,
where the only zero terms of these sequences are the first $m-1$ terms.
Define: $\mathbb{R}^{\infty} \times \mathbb{R}^{\infty} \rightarrow \mathbb{R}_{++}$ by $D((x_m),(y_m)):=\text{sup}_{m \in \mathbb{N}}\text{min}\{1, d(x_m, y_m)\}$, and check that $D$ is a metric on $\mathbb{R}^{\infty}$. Do $x_m$, and/or $y_m$ converge in $\mathbb{R}^{\infty}$ and in $(\mathbb{R}^{\infty},D)$?
It seems obvious that $x_m$ diverge and $y_m$ converges to zero in $\mathbb{R}^{\infty}$.
I am not sure about the next steps. In $(\mathbb{R}^{\infty},D)$, $y_m$ would converge to $\{0,0,0,...\}$. And I think $x_m$ would diverge in $(\mathbb{R}^{\infty},D)$. Consider $z=\{0,0,...0, n, n, n,...\}$. Whatever number $n$ would be, there always exists a number $M \in \mathbb{N}$ such that $D$ is greater than zero. Am I correct?