straight line is shortest curve between two points

Suppose $p$ and $q$ are two distinct points in ${ℝ}^n$,and $\gamma$ is a rectifiable curve from $p$ to $q$.Then every curve other than the straight line segment from $p$ to $q$has a length greater than the Euclidean distance $\parallel p-q\parallel$.

Proof.

Let $\gamma :[0,1]\to {ℝ}^n$be the curve with length $L$.If it is not straight, then there exists a point $x=\gamma (t)$ that does not lie on the line segment from $p$ to $q$.We have

$$L\ge \parallel q-x\parallel +\parallel x-p\parallel >\parallel p-q\parallel .$$

The first inequality comes from the definition of $L$ as the least upper bound of the length of any broken-line approximation to the curve $\gamma$.The second inequality is the usual triangle inequality,but it is a strict inequality since $x$ lies outside the line segment between $p$ and $q$,as shown in the following diagram.∎