How Does the Triangle Inequality Transform from Equality to Inequality?

[Bashyboy]

Hello all,

I am currently reading about the triangle inequality, from this article
http://people.sju.edu/~pklingsb/cs.triang.pdf

I am curious, how does the equality transform into an inequality? Does it take on this change because one takes the absolute value of 2uv? Because before the absolute value, 2uv could be a negative value, thus making all of |u|^2 + 2uv + |v|^2 smaller, is this correct?

 

[UltrafastPED]

You are correct ... that is why the first inequality appears. The second one is from the Cauchy-Schwartz inequality, as noted.

These are properties that are required for a metric space.

 

[Bashyboy]

I have one other question. In the article, it says that since both sides of the inequality of non-negative, it is permissible to then square both sides of the inequality. Why would it not be possible to square both sides if both sides were negative?

 

[UltrafastPED]

I'm sure that they said "you can square each term since they are all positive". Try that with this inequality:

1 - 2 < 1 ... hence the requirement for all positive.