Triangle Inequality Proof

  • Thread starter Bashyboy
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  • #1
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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?
 

Answers and Replies

  • #2
UltrafastPED
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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.
 
  • #3
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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?
 
  • #4
UltrafastPED
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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.
 

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