Can I Simplify Summation Notation in Proving the Triangle Inequality?

[tylerc1991]

Homework Statement

My first post on PF and just a quick question about summation notation usage: given the same start and end values for all sums used, can I just 'remove', so to speak, the sum signs? For example, I'm trying to prove the triangle inequality and have the equation narrowed down to:

\sum|x-z| <= \sum|x-y|+|y-z|

now can I simply remove the sigma's to yield:

|x-z| <= |x-y|+|y-z| ?

I know this seems like an odd question but I know I've seen this operation somewhere in the past and was unsure of it's correct usage. Thanks a ton!

Homework Equations

The Attempt at a Solution

 

[gomunkul51]

LaTeX Code: \\sum |x-z| <= LaTeX Code: \\sum |x-y|+|y-z|

This is ambiguous.

where do the summations start and where they end? what are the indexes?
As a rule: if they are from 0 to N (any integer N) they you can just pick N to be 1 and what you are doing is correct. if the summation is from 0 to infinity on the other hand this is not always true.

for your particular question it might be true, you need to do some algebraic manipulation to prove it is so, it is very similar to the triangle inequality...

 

[tylerc1991]

OK that makes sense. See I am trying to prove that a function is a metric (the taxicab metric to be precise). As I said, the start and end values are all the same for each sum (1 to n for R^n). I didn't know how to put the indexes in so I just stated them in words. So for my purpose, I could just set n = 1 and the proof would be done?