Metric spaces - $(0, 1)$ and $\mathbb{R}$ are not isometric

Question

I'm looking for alternative solutions to what I currently have for the sake of self studying to the following:

Show that $(0,1)$ and $\mathbb{R}$ are not isometric, where both sets are equipped with the standard $d_2$ metric.

Current solution:

$d(0, 2) = 2$ in $\mathbb{R}$, but $d(f(0), f(2))$ is at most $1$ in $(0,1)$.

Answer 1

Isometries preserve boundedness, completeness and total boundedness, among other things.

$(0,1)$ is totally bounded and bounded, the reals are not.

$(0,1)$ is not complete ($\frac{1}{n}$ is Cauchy but does not converge) and the reals are.