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.
This is just by looking at all the cases.
The triangle inequality in
which also has an
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 -
which can also be useful, especially if we don't know if