How does $\mathbb R \subseteq B(0,2)$ hold in the discrete metric? How is this hold that $\mathbb R \subseteq B(0,2)$ where 
$\big<\mathbb R,d\big>$ and d is a discrete metric?
By doing so we showed that $\mathbb R $ is bounded

Mimicking my answer to your previous question:
Recall that
$$
B(0,2) = \{x\in\mathbb{R} \mid d(0,x)<2\}.
$$
If $d$ is the discrete metric, then
$$
d(0,x)=\begin{cases}
0 & \text{if $x=0$}\\
1 & \text{otherwise}
\end{cases}
$$ so that $d(0,x)<2$ for every $x\in\mathbb{R}$. Therefore $B(0,2)=\mathbb{R}$, so that $\mathbb{R}$ is bounded in the discrete metric.
