# How to prove the following using triangle inequality?

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?

• 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. Sep 14 '20 at 2:54
• 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? Sep 14 '20 at 2:56
• @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. 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.