Problem: Show that for any $a, b \in ℝ ^n: \|a + b\|\geq\|a\|-\|b\|$

I have a feeling that we can use the triangle inequality here somehow. I am not sure how to start this proof?

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    $\begingroup$ Do not post unsearchable images of text. Instead typeset using MathJax. Oh... and just expand the left-hand-side of the inequality. The result drops out immediately. $\endgroup$ Sep 14 '20 at 2:54
  • $\begingroup$ As for how to continue... you are correct to try to think of the triangle inequality. Perhaps if we were to rearrange things a bit, $\|a\|\leq \|a+b\|+\|b\|$ this might be a bit more recognizable to you? What if we were to remind you that $0 = b-b$ and that anything plus "zero" is equal to itself? $\endgroup$
    – JMoravitz
    Sep 14 '20 at 2:56
  • $\begingroup$ @DavidG.Stork "expand the left side", most would think you mean as $\|a+b\|\leq \|a\|+\|b\|$. That tells us things that $\|a+b\|$ is less than, but not much about what it is greater than. $\endgroup$
    – JMoravitz
    Sep 14 '20 at 2:57

Note that $(a+b)+(-b)=a$. Therefore by the triangle inequality as you suggest, $$\|a\|=\|(a+b)+(-b)\|\leq\|a+b\|+\|-b\|=\|a+b\|+\|b\|.$$ Rearranging gives your desired result.


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