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I have to prove: $$|z|\leq|Re(z)|+|Im(z)|$$

Work done so far: I know how to prove the triangle inequality for two vectors but I am not sure how to show it holds for two components of a single vector. Any help would be appreciated.

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    $\begingroup$ Consider the vectors $(x,y)$, $(x,0)$ and $(0,y)$. $\endgroup$ – Robert Z Jan 24 '19 at 6:09
  • $\begingroup$ @RobertZ You are right, thanks! $\endgroup$ – Bertrand Wittgenstein's Ghost Jan 24 '19 at 6:15
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Note that $|z| = \sqrt{|Re(z)|^2+|Im(z)|^2}$, and the inequality follows immediately after squaring both sides.

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Use $z=\text{Re }z+i\,\text{Im }z$ and the conventional triangle inequality: $$|z|\le|\text{Re }z|+|i\,\text{Im }z|=|\text{Re }z|+|\text{Im }z|.$$

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