Transitive closure of $H=\{(a,b) \in \mathbb{R}^2: |a-b| \leq 0.1\}$

Question

$$H = \{(a, b) \in \mathbb{R}^2: |a − b| \leq 0.1\}$$

In class today we went over this problem as an example to show transitive closure. I know that the transitive closure of $H$ is "All real numbers", but I am not sure why. Would someone here be able to explain it to me?

Answer 1

If $|a-b| < n \cdot 0.1$, then $(a,b)$ lies in $H^{\circ n}$ and hence in the transitive closure. Since $\{n \cdot 0.1 : n \in \mathbb{N}\}$ is unbounded, we are done.