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.

Given real numbers

The triangle inequality.

For all

**Proof.**

This is just by looking at all the cases.

**Subproof.**

**Case 1:**

**Subproof.**

*Case 1a:*

**Subproof.**

*Case 1b:*

**Subproof.**

*Case 1c:*

Also:

**Subproof.**

**Case 2:**

**Subproof.**

*Case 2a:*

**Subproof.**

*Case 2b:*

**Subproof.**

*Case 2c:*

The triangle inequality in

Note the

which also has an intermediate

point

We can also write the triangle inequality in

To derive this from the other versions just note that

Hence the result. Its alternative form,

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

and

which can also be useful, especially if we don't know if