Editing Triangle inequality (section)

====Relationship with shortest paths====
[[File:Arclength.svg|300px|thumb|The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.]]
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

No polygonal path between two points is shorter than the line between them. This implies that no curve can have an [[arc length]] less than the distance between its endpoints. By definition, the arc length of a curve is the [[least upper bound]] of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.<ref>{{cite book|title=Numbers and Geometry|author=John Stillwell|author-link=John Stillwell|year=1997|publisher=Springer|isbn=978-0-387-98289-2|url=https://books.google.com/books?id=4elkHwVS0eUC&pg=PA95}} p. 95.</ref>