I What Are the Benefits of Triangle Inequality in Mathematics?

[MiddleEast]

Hi,
Recently I studied triangle inequality and the proof using textbook precalculus by David Cohen.
My question is whats the benefit of this inequality ? One benefit I found is to solve inequality of the form |x+a| + |x+b| < c which make the solution much easier than taking cases. I assume this inequality can be used in proof? the beauty of this inequality is to separate absolute of sum to sum of absolutes which - supposedly - will make proving (whatever the proof is) much easier.

Are there any other benefits ?
Are there any important inequality other triangle and AM-GM inequality that quite famous ?
Thanks.

 

[PeroK]

The triangle inequality is a fundamental defining property of a distance function or metric (of which $| x - y|$ is probably the first you'll encounter). If you have a set and you want to have a notion of the distance between two elements of that set, which we'll denote by $d(x, y)$, then we have four fundamental properties. Here $x, y, z$ are any elements in your set.
$$\text{1)} \ d(x, y) \ge 0$$$$\text{2)} \ d(x, y) = 0 \ \Leftrightarrow \ x = y$$$$\text{3)} \ d(x, y) = d(y, x)$$$$\text{4)} \ \text{(the triangle inequality)} \ d(x, z) \le d(x, y) + d(y, z)$$
In any case, the triangle inequality is used all over mathematics and physics.

 

[MidgetDwarf]

Hi,
Recently I studied triangle inequality and the proof using textbook precalculus by David Cohen.
My question is whats the benefit of this inequality ? One benefit I found is to solve inequality of the form |x+a| + |x+b| < c which make the solution much easier than taking cases. I assume this inequality can be used in proof? the beauty of this inequality is to separate absolute of sum to sum of absolutes which - supposedly - will make proving (whatever the proof is) much easier.

Are there any other benefits ?
Are there any important inequality other triangle and AM-GM inequality that quite famous ?
Thanks.

As Perok mentioned, thats the idea of the triangle inequality. It is also a useful tool for proving properties of limits of sequences and functions in Analysis, Topologies with a metric...