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I'm trying to understand the triangle inequality theorem and part of proof contains $$2\textrm{Re}(\left) \leqq 2\left|\left\right|.$$ Can someone show me how this is so?

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    Would [math.se] be a better home for this question? – 2017-02-15

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The result of the inner product $\left$ is a complex number, so let's write it as $c+id$. $$\textrm{Re}(c+id) = c$$ $$\left|c+id\right| = \sqrt{c^2 + d^2}$$