Is this proof correct? Establishing that $|x+y| geq | |x| - |y| |$












3












$begingroup$


Using that $|a|+|b|geq|a+b|$
begin{align}
|-x|+|x+y| geq |-x+x+y| = |y|\
|-y|+|x+y| geq |-y+x+y| = |x|
end{align}



Substracting $|-x|$ from the first inequality and $|-y|$ from the second:



begin{align}
|x+y| geq |y|-|-x| \
|x+y| geq |x| - |-y|
end{align}



Using the fact that if $a geq b$ and $a geq -b$ then $a geq |b|$.



begin{align}
|x+y| geq ||x| - |-y|| = ||x| - |y||
end{align}



Is the above correct?






real-analysis elementary-number-theory inequality







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    The proof is good!
    $endgroup$
    – Theo Bendit
    Dec 18 '18 at 23:34






  • 1




    $begingroup$
    Seems good enough!
    $endgroup$
    – Rebellos
    Dec 18 '18 at 23:40






  • 1




    $begingroup$
    Nice work. See also Help checking proof of $|x| - |y| leq |x+y|$
    $endgroup$
    – amWhy
    Dec 18 '18 at 23:57












  • $begingroup$
    Perfect. But I would have typed $|(|x|-|y|)|$ or $|; |x|-|y|;|$(etc.) to make it easier to read, and to avoid confusion with the functional-analysis symbol $||z||$ (also written $|z|,$ coded as |z|). You can use ; and , to add a little space between key-strokes.
    $endgroup$
    – DanielWainfleet
    Dec 19 '18 at 3:16


















3












$begingroup$


Using that $|a|+|b|geq|a+b|$
begin{align}
|-x|+|x+y| geq |-x+x+y| = |y|\
|-y|+|x+y| geq |-y+x+y| = |x|
end{align}



Substracting $|-x|$ from the first inequality and $|-y|$ from the second:



begin{align}
|x+y| geq |y|-|-x| \
|x+y| geq |x| - |-y|
end{align}



Using the fact that if $a geq b$ and $a geq -b$ then $a geq |b|$.



begin{align}
|x+y| geq ||x| - |-y|| = ||x| - |y||
end{align}



Is the above correct?






real-analysis elementary-number-theory inequality







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    The proof is good!
    $endgroup$
    – Theo Bendit
    Dec 18 '18 at 23:34






  • 1




    $begingroup$
    Seems good enough!
    $endgroup$
    – Rebellos
    Dec 18 '18 at 23:40






  • 1




    $begingroup$
    Nice work. See also Help checking proof of $|x| - |y| leq |x+y|$
    $endgroup$
    – amWhy
    Dec 18 '18 at 23:57












  • $begingroup$
    Perfect. But I would have typed $|(|x|-|y|)|$ or $|; |x|-|y|;|$(etc.) to make it easier to read, and to avoid confusion with the functional-analysis symbol $||z||$ (also written $|z|,$ coded as |z|). You can use ; and , to add a little space between key-strokes.
    $endgroup$
    – DanielWainfleet
    Dec 19 '18 at 3:16
















3












3








3





$begingroup$


Using that $|a|+|b|geq|a+b|$
begin{align}
|-x|+|x+y| geq |-x+x+y| = |y|\
|-y|+|x+y| geq |-y+x+y| = |x|
end{align}



Substracting $|-x|$ from the first inequality and $|-y|$ from the second:



begin{align}
|x+y| geq |y|-|-x| \
|x+y| geq |x| - |-y|
end{align}



Using the fact that if $a geq b$ and $a geq -b$ then $a geq |b|$.



begin{align}
|x+y| geq ||x| - |-y|| = ||x| - |y||
end{align}



Is the above correct?






real-analysis elementary-number-theory inequality







share|cite|improve this question









$endgroup$




Using that $|a|+|b|geq|a+b|$
begin{align}
|-x|+|x+y| geq |-x+x+y| = |y|\
|-y|+|x+y| geq |-y+x+y| = |x|
end{align}



Substracting $|-x|$ from the first inequality and $|-y|$ from the second:



begin{align}
|x+y| geq |y|-|-x| \
|x+y| geq |x| - |-y|
end{align}



Using the fact that if $a geq b$ and $a geq -b$ then $a geq |b|$.



begin{align}
|x+y| geq ||x| - |-y|| = ||x| - |y||
end{align}



Is the above correct?






real-analysis elementary-number-theory inequality




real-analysis elementary-number-theory inequality






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 18 '18 at 23:26









PintecoPinteco

731313




731313








  • 3




    $begingroup$
    The proof is good!
    $endgroup$
    – Theo Bendit
    Dec 18 '18 at 23:34






  • 1




    $begingroup$
    Seems good enough!
    $endgroup$
    – Rebellos
    Dec 18 '18 at 23:40






  • 1




    $begingroup$
    Nice work. See also Help checking proof of $|x| - |y| leq |x+y|$
    $endgroup$
    – amWhy
    Dec 18 '18 at 23:57












  • $begingroup$
    Perfect. But I would have typed $|(|x|-|y|)|$ or $|; |x|-|y|;|$(etc.) to make it easier to read, and to avoid confusion with the functional-analysis symbol $||z||$ (also written $|z|,$ coded as |z|). You can use ; and , to add a little space between key-strokes.
    $endgroup$
    – DanielWainfleet
    Dec 19 '18 at 3:16
















  • 3




    $begingroup$
    The proof is good!
    $endgroup$
    – Theo Bendit
    Dec 18 '18 at 23:34






  • 1




    $begingroup$
    Seems good enough!
    $endgroup$
    – Rebellos
    Dec 18 '18 at 23:40






  • 1




    $begingroup$
    Nice work. See also Help checking proof of $|x| - |y| leq |x+y|$
    $endgroup$
    – amWhy
    Dec 18 '18 at 23:57












  • $begingroup$
    Perfect. But I would have typed $|(|x|-|y|)|$ or $|; |x|-|y|;|$(etc.) to make it easier to read, and to avoid confusion with the functional-analysis symbol $||z||$ (also written $|z|,$ coded as |z|). You can use ; and , to add a little space between key-strokes.
    $endgroup$
    – DanielWainfleet
    Dec 19 '18 at 3:16










3




3




$begingroup$
The proof is good!
$endgroup$
– Theo Bendit
Dec 18 '18 at 23:34




$begingroup$
The proof is good!
$endgroup$
– Theo Bendit
Dec 18 '18 at 23:34




1




1




$begingroup$
Seems good enough!
$endgroup$
– Rebellos
Dec 18 '18 at 23:40




$begingroup$
Seems good enough!
$endgroup$
– Rebellos
Dec 18 '18 at 23:40




1




1




$begingroup$
Nice work. See also Help checking proof of $|x| - |y| leq |x+y|$
$endgroup$
– amWhy
Dec 18 '18 at 23:57






$begingroup$
Nice work. See also Help checking proof of $|x| - |y| leq |x+y|$
$endgroup$
– amWhy
Dec 18 '18 at 23:57














$begingroup$
Perfect. But I would have typed $|(|x|-|y|)|$ or $|; |x|-|y|;|$(etc.) to make it easier to read, and to avoid confusion with the functional-analysis symbol $||z||$ (also written $|z|,$ coded as |z|). You can use ; and , to add a little space between key-strokes.
$endgroup$
– DanielWainfleet
Dec 19 '18 at 3:16






$begingroup$
Perfect. But I would have typed $|(|x|-|y|)|$ or $|; |x|-|y|;|$(etc.) to make it easier to read, and to avoid confusion with the functional-analysis symbol $||z||$ (also written $|z|,$ coded as |z|). You can use ; and , to add a little space between key-strokes.
$endgroup$
– DanielWainfleet
Dec 19 '18 at 3:16












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