Isomorphism to Closed Interval

Theorem

Let $m, n \in \N$ such that $m < n$.

Then $\left|{\left[{m + 1 .. n}\right]}\right| = n - m$.

Let $h: \N_{n - m} \to \left[{m + 1 .. n}\right]$ be the mapping defined as:

$\forall x \in \N_{n - m}: h \left({x}\right) = x + m + 1$

Then $h$ is the unique isomorphism as defined in Unique Isomorphism between Finite Totally Ordered Sets, where the orderings on $\left[{m + 1 .. n}\right]$ and $\N_{n - m}$ are those induced by the ordering of $\N$.

Sources

• Seth Warner: Modern Algebra (1965): $\S 17$: Theorem $17.10$