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?

  • $\begingroup$ The question that presents itself : what can you infer from this inequality ? $\endgroup$ Aug 17, 2018 at 6:54
  • $\begingroup$ Search for triangle inequality. $\endgroup$ Aug 17, 2018 at 7:00

1 Answer 1


$||x||=||x-y+y|| \le ||x-y||+||y||$, 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.

  • $\begingroup$ Oh.............Thank you for your answer.... $\endgroup$
    – Keith
    Aug 17, 2018 at 6:52
  • $\begingroup$ Why the downvote ?????????????????????????????? $\endgroup$
    – Fred
    Aug 17, 2018 at 7:14
  • 1
    $\begingroup$ 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. $\endgroup$ Aug 17, 2018 at 10:16
  • $\begingroup$ Yes, my sheriff ! $\endgroup$
    – Fred
    Aug 17, 2018 at 10:20

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