TITLE: Prove that $[0,1]$ is not isometric to $[0,2]$.
QUESTION [6 upvotes]: Prove that $[0,1]$ is not isometric to $[0,2]$.

Suppose there is an isometry $f:[0,1]\to[0,2]$. Since f is continuous and surjective, the only values for $f(0)$ and $f(1)$ are $f(0)=0$ and $f(1)=2$, or $f(0)=2$ and $f(1)=0$. In either case, $|f(1)-f(0)|=2$. This contradicts $f$ being distance-preserving.
Is this right?

REPLY [2 votes]: The diameter of a metric space is the supremum, taken over all pairs of points in the space, of the distance between the points of the pair. For a compact metric space the diameter is finite and is attained for some pair of points in the space. Isometric metric spaces have the same diameter. The spaces $[0, 1]$ and $[0, 2]$ with the Eucliden metric have the distinct diameters $1$ and $2$ (obviously), so are not isometric.