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
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.
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
This is just by looking at all the cases.
Case 1: . Then . There are three subcases. They are all very similar.
Case 1a: , so and giving .
Case 1b: , so and giving .
Case 1c: , so and giving .
Case 2: . Then . There are three subcases.
Case 2a: , so and giving .
Case 2b: , so and giving .
Case 2c: , so 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
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
Hence the result. Its alternative form,
can be derived in a similar way. If we switch for we can even get
which can also be useful, especially if we don't know if is negative or positive.