# Triangle Inequality Proof

Hello all,

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
Gold Member
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.

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