Proving $mid parallel x parallel - parallel y parallel mid$ $leqq$ $parallel x-y parallel$? [duplicate]

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  • Reverse Triangle Inequality Proof

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Let $V$ be a normed vector space and I need to prove the follwing inequality $mid parallel x parallel - parallel y parallel mid$ $leqq$ $parallel x-y parallel$ containing the norm and the absolute value of the real numbers. However things just get twisted and I cannot see how to prove it.... Could anyone please tell me how to prove it?







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marked as duplicate by Jyrki Lahtonen, Martin R, Nosrati, Arnaud D., amWhy Aug 17 at 11:13


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • The question that presents itself : what can you infer from this inequality ?
    – Neil hawking
    Aug 17 at 6:54










  • Search for triangle inequality.
    – Jyrki Lahtonen
    Aug 17 at 7:00














up vote
-2
down vote

favorite













This question already has an answer here:



  • Reverse Triangle Inequality Proof

    3 answers



Let $V$ be a normed vector space and I need to prove the follwing inequality $mid parallel x parallel - parallel y parallel mid$ $leqq$ $parallel x-y parallel$ containing the norm and the absolute value of the real numbers. However things just get twisted and I cannot see how to prove it.... Could anyone please tell me how to prove it?







share|cite|improve this question












marked as duplicate by Jyrki Lahtonen, Martin R, Nosrati, Arnaud D., amWhy Aug 17 at 11:13


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • The question that presents itself : what can you infer from this inequality ?
    – Neil hawking
    Aug 17 at 6:54










  • Search for triangle inequality.
    – Jyrki Lahtonen
    Aug 17 at 7:00












up vote
-2
down vote

favorite









up vote
-2
down vote

favorite












This question already has an answer here:



  • Reverse Triangle Inequality Proof

    3 answers



Let $V$ be a normed vector space and I need to prove the follwing inequality $mid parallel x parallel - parallel y parallel mid$ $leqq$ $parallel x-y parallel$ containing the norm and the absolute value of the real numbers. However things just get twisted and I cannot see how to prove it.... Could anyone please tell me how to prove it?







share|cite|improve this question













This question already has an answer here:



  • Reverse Triangle Inequality Proof

    3 answers



Let $V$ be a normed vector space and I need to prove the follwing inequality $mid parallel x parallel - parallel y parallel mid$ $leqq$ $parallel x-y parallel$ containing the norm and the absolute value of the real numbers. However things just get twisted and I cannot see how to prove it.... Could anyone please tell me how to prove it?





This question already has an answer here:



  • Reverse Triangle Inequality Proof

    3 answers









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share|cite|improve this question




share|cite|improve this question










asked Aug 17 at 6:44









Keith

1,350720




1,350720




marked as duplicate by Jyrki Lahtonen, Martin R, Nosrati, Arnaud D., amWhy Aug 17 at 11:13


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






marked as duplicate by Jyrki Lahtonen, Martin R, Nosrati, Arnaud D., amWhy Aug 17 at 11:13


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • The question that presents itself : what can you infer from this inequality ?
    – Neil hawking
    Aug 17 at 6:54










  • Search for triangle inequality.
    – Jyrki Lahtonen
    Aug 17 at 7:00
















  • The question that presents itself : what can you infer from this inequality ?
    – Neil hawking
    Aug 17 at 6:54










  • Search for triangle inequality.
    – Jyrki Lahtonen
    Aug 17 at 7:00















The question that presents itself : what can you infer from this inequality ?
– Neil hawking
Aug 17 at 6:54




The question that presents itself : what can you infer from this inequality ?
– Neil hawking
Aug 17 at 6:54












Search for triangle inequality.
– Jyrki Lahtonen
Aug 17 at 7:00




Search for triangle inequality.
– Jyrki Lahtonen
Aug 17 at 7:00










1 Answer
1






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up vote
1
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accepted










$||x||=||x-y+y|| le ||x-y||+||y||$, hence



$(1)$ $||x||-||y|| le ||x-y||$.



In a similar way we get



$(2)$ $||y||-||x|| le ||y-x||$.



Since $||y-x||=||x-y||$, $(1)$ and $(2)$ give the result.






share|cite|improve this answer




















  • Oh.............Thank you for your answer....
    – Keith
    Aug 17 at 6:52










  • Why the downvote ??????????????????????????????
    – Fred
    Aug 17 at 7:14






  • 1




    The downvote is because it looks like it didn't occur to you to that may be, during its 8 years of existence, the site has already handled the triangle inequality. IMO it is not useful to repeat material already explained multiples of times. And "downvote" = "this answer is not useful" right click it to see the explanation.
    – Jyrki Lahtonen
    Aug 17 at 10:16











  • Yes, my sheriff !
    – Fred
    Aug 17 at 10:20

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










$||x||=||x-y+y|| le ||x-y||+||y||$, hence



$(1)$ $||x||-||y|| le ||x-y||$.



In a similar way we get



$(2)$ $||y||-||x|| le ||y-x||$.



Since $||y-x||=||x-y||$, $(1)$ and $(2)$ give the result.






share|cite|improve this answer




















  • Oh.............Thank you for your answer....
    – Keith
    Aug 17 at 6:52










  • Why the downvote ??????????????????????????????
    – Fred
    Aug 17 at 7:14






  • 1




    The downvote is because it looks like it didn't occur to you to that may be, during its 8 years of existence, the site has already handled the triangle inequality. IMO it is not useful to repeat material already explained multiples of times. And "downvote" = "this answer is not useful" right click it to see the explanation.
    – Jyrki Lahtonen
    Aug 17 at 10:16











  • Yes, my sheriff !
    – Fred
    Aug 17 at 10:20














up vote
1
down vote



accepted










$||x||=||x-y+y|| le ||x-y||+||y||$, hence



$(1)$ $||x||-||y|| le ||x-y||$.



In a similar way we get



$(2)$ $||y||-||x|| le ||y-x||$.



Since $||y-x||=||x-y||$, $(1)$ and $(2)$ give the result.






share|cite|improve this answer




















  • Oh.............Thank you for your answer....
    – Keith
    Aug 17 at 6:52










  • Why the downvote ??????????????????????????????
    – Fred
    Aug 17 at 7:14






  • 1




    The downvote is because it looks like it didn't occur to you to that may be, during its 8 years of existence, the site has already handled the triangle inequality. IMO it is not useful to repeat material already explained multiples of times. And "downvote" = "this answer is not useful" right click it to see the explanation.
    – Jyrki Lahtonen
    Aug 17 at 10:16











  • Yes, my sheriff !
    – Fred
    Aug 17 at 10:20












up vote
1
down vote



accepted







up vote
1
down vote



accepted






$||x||=||x-y+y|| le ||x-y||+||y||$, hence



$(1)$ $||x||-||y|| le ||x-y||$.



In a similar way we get



$(2)$ $||y||-||x|| le ||y-x||$.



Since $||y-x||=||x-y||$, $(1)$ and $(2)$ give the result.






share|cite|improve this answer












$||x||=||x-y+y|| le ||x-y||+||y||$, hence



$(1)$ $||x||-||y|| le ||x-y||$.



In a similar way we get



$(2)$ $||y||-||x|| le ||y-x||$.



Since $||y-x||=||x-y||$, $(1)$ and $(2)$ give the result.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Aug 17 at 6:51









Fred

38.2k1238




38.2k1238











  • Oh.............Thank you for your answer....
    – Keith
    Aug 17 at 6:52










  • Why the downvote ??????????????????????????????
    – Fred
    Aug 17 at 7:14






  • 1




    The downvote is because it looks like it didn't occur to you to that may be, during its 8 years of existence, the site has already handled the triangle inequality. IMO it is not useful to repeat material already explained multiples of times. And "downvote" = "this answer is not useful" right click it to see the explanation.
    – Jyrki Lahtonen
    Aug 17 at 10:16











  • Yes, my sheriff !
    – Fred
    Aug 17 at 10:20
















  • Oh.............Thank you for your answer....
    – Keith
    Aug 17 at 6:52










  • Why the downvote ??????????????????????????????
    – Fred
    Aug 17 at 7:14






  • 1




    The downvote is because it looks like it didn't occur to you to that may be, during its 8 years of existence, the site has already handled the triangle inequality. IMO it is not useful to repeat material already explained multiples of times. And "downvote" = "this answer is not useful" right click it to see the explanation.
    – Jyrki Lahtonen
    Aug 17 at 10:16











  • Yes, my sheriff !
    – Fred
    Aug 17 at 10:20















Oh.............Thank you for your answer....
– Keith
Aug 17 at 6:52




Oh.............Thank you for your answer....
– Keith
Aug 17 at 6:52












Why the downvote ??????????????????????????????
– Fred
Aug 17 at 7:14




Why the downvote ??????????????????????????????
– Fred
Aug 17 at 7:14




1




1




The downvote is because it looks like it didn't occur to you to that may be, during its 8 years of existence, the site has already handled the triangle inequality. IMO it is not useful to repeat material already explained multiples of times. And "downvote" = "this answer is not useful" right click it to see the explanation.
– Jyrki Lahtonen
Aug 17 at 10:16





The downvote is because it looks like it didn't occur to you to that may be, during its 8 years of existence, the site has already handled the triangle inequality. IMO it is not useful to repeat material already explained multiples of times. And "downvote" = "this answer is not useful" right click it to see the explanation.
– Jyrki Lahtonen
Aug 17 at 10:16













Yes, my sheriff !
– Fred
Aug 17 at 10:20




Yes, my sheriff !
– Fred
Aug 17 at 10:20


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