Where Does the Less Than Symbol Disappear in the Triangle Inequality Proof?

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SUMMARY

The discussion centers on the Triangle Inequality Proof, specifically the transition from the inequality |x+y| ≤ |x| + |y| to the squared form (|x+y|)^2 ≤ (|x| + |y|)^2. The less than symbol disappears as the proof progresses through algebraic manipulation, ultimately demonstrating that 2xy ≤ 2|x||y|. This conclusion is established by recognizing that x ≤ |x| and y ≤ |y|, confirming the validity of the inequality.

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Punkyc7
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Im curios as to why the inquality is
||x+y||[tex]\leq[/tex]||X||+||y||

but the end of the proof is

=(||x||+||Y||)^2

where does the less than symbol disappear too
 
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[tex]|x+y|\leq |x|+|y|\Leftrightarrow(|x+y|)^2\leq (|x|+|y|)^2\Leftrightarrow(x+y)^2\leq (|x|+|y|)^2\Leftrightarrow x^2+2xy+y^2\leq|x|^2+2|x||y|+|y|^2\Leftrightarrow x^2+2xy+y^2\leq x^2+2|x||y|+y^2\Leftrightarrow 2xy\leq2|x||y|[/tex]
[tex]xy\leq|x||y|[/tex]

That is obvious since

[tex]x\leq|x|[/tex] and [tex]y\leq|y|[/tex]
 
Last edited:

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