(Theorem) The countable product of the real line is metrizable

Theorem (Munkres 20.5): ${\mathbb{R}}^{\omega }$ is metrizable

Let $\overline{d}(a,b)=\text{min}\{|a-b|,1\}$ be the standard bounded metric on $\mathbb{R}$. If $\mathbf{x},\mathbf{y}$ are two sequences in ${\mathbb{R}}^{\omega }$ (The countably infinite product of the real line with itself), define

$$D(x,y)=\text{sup}\left\{\frac{\overline{d}(x_i,y_i)}{i}\right\},$$

where $i>0$. Then (1) $D$ is a metric on ${\mathbb{R}}^{\omega }$ and (2) $D$ induces the product topology.