Struggling to prove simple triangle inequality

• To: mathgroup at smc.vnet.net
• Subject: [mg126311] Struggling to prove simple triangle inequality
• Date: Mon, 30 Apr 2012 04:42:19 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Greetings!

Given two 3D vectors A and B, I want to prove that length of their sum
is less or equal than the sum of their lengths:

length[v_] := Sqrt[v.v];
a = {ax, ay, az};
b = {bx, by, bz};
inequality = length[a + b] <= length[a] + length[b];

This is famous, well-known and quite obvious: triangle side is shorter
than the sum of other sides, straight line is shorter than non-
straight, etc. However, proving it formally is hard.

This fails:

assum = Element[ax | ay | az | bx | by | bz, Reals];
FullSimplify[inequality, assum]

This takes ages on a high-end PC with unknown result:

vars = {ax, ay, az, bx, by, bz};
Reduce[inequality, vars, Reals]

Anyone can help? I think Reduce should somehow make it because it
actually succeeds at least with 2D vectors.

--
All the best,