Definition 3

The length of a bounded interval with end-points a and b (a \leq b) is defined to be b - a.

Every bounded interval which contains more than one point therefore has length > 0. If $a \leq b$, the four intervals $[a, b], ]a, b[, [a, b[$ and $]a, b[$ all have the same length. An interval with end-points $a + c$ and $b + c$ has the same length as an interval with end-points $a$ and $b$; in other words, the length of an interval is invariant under translation.

If $a \leq c \leq d \leq b$ we have $d - c \leq b - a$. Hence if a bounded interval $I$ is contained in a bounded interval $I'$, the length of $I$ is less than or equal to the length of $I'$.

If $n$ mutually disjoint open intervals $I_1, I_2, \ldots, I_n$ are contained in the interval $[a, b]$ ($a < b$) it is easily seen by induction on $n$ that, if $I_k = ]c_k, d_k[$, there is a permutation $\sigma$ of the indices $k$ ($1 \leq k \leq n$) such that $d_{\sigma(k)} \leq c_{\sigma(k+1)}$ for $1 \leq k \leq n - 1$. It follows immediately that the sum of the lengths of the intervals $I_k$ is at most equal to the length of $[a, b]$, and that equality holds if and only if $c_{\sigma(1)} = a, d_{\sigma(n)} = b$ and $d_{\sigma(k)} = c_{\sigma(k+1)}$ for $1 \leq k \leq n - 1$.