The triangle inequality

1. Introduction

The single most important inequality in analysis is the triangle inequality, and it will be used a lot throughout this course. Later on it becomes the main building block for a more general theory of analysis that you learn about when you study metric spaces.

The triangle inequality concerns distance between points and says that the straight line distance between and is less than the sum of the distances from to and from to . It is very much part of our everyday intuition about distances and easy to remember. It is, however, very useful.

2. Distances in the reals

Given real numbers and the value - represents the distance along the numberline from to . By definition, it is equal to - if and to - if > . So - = - . We will sometimes denote this distance by (,) or (,) .

The triangle inequality.

For all ,, we have

(,) (,)+(,)

Proof.

This is just by looking at all the cases.

Subproof.

Case 1: . Then (,)= - =- . There are three subcases. They are all very similar.

Subproof.

Case 1a: , so (,)= - =- 0 and (,)= - =- -=(,) giving (,)+(,) (,) .

Subproof.

Case 1b: > , so (,)= - =- and (,)= - =- giving (,)+(,)=-+-=-=(,) .

Subproof.

Case 1c: > , so (,)= - =- - and (,)= - =- 0 giving (,)+(,)=-=(,) .

Also:

Subproof.

Case 2: < . Then (,)= - =- . There are three subcases.

Subproof.

Case 2a: , so (,)= - =- (,) and (,)= - =- 0 giving (,)+(,) (,) .

Subproof.

Case 2b: > , so (,)= - =- and (,)= - =- giving (,)+(,)=-+-=-=(,) .

Subproof.

Case 2c: > , so (,)= - =- 0 and (,)= - =- - giving (,)+(,)=-=(,) .

The triangle inequality in takes the form

- - + -

Note the , the + sign and the introduced intermediate point on the right. By rearranging we have - - - - , or

- - - -

which also has an intermediate point added on the right in the same way, but the has changed to and the + has changed to -. Both forms are equally useful. I find the first easy to remember; the hints here will help you remember the second just as easily.

We can also write the triangle inequality in in the form

+ +

To derive this from the other versions just note that

-(-) -0 +0-(-)

Hence the result. Its alternative form,

+ -

can be derived in a similar way. If we switch - for we can even get

- +

and

+ -

which can also be useful, especially if we don't know if is negative or positive.