Struggling to prove simple triangle inequality

*To*: mathgroup at smc.vnet.net*Subject*: [mg126311] Struggling to prove simple triangle inequality*From*: Vladimir M <vladimir7523 at gmail.com>*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, Vladimir