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Show that if $|arg(z)| \leq \frac{\pi}{4}$ then $ x \geq 0 $ and $|z| \leq \sqrt{2}x$, where $z=x+iy$.

My question is : How can I prove that x is greater than 0 and the other is, can I use the fact tangent function is increasing function?, for which interval of real numbers in tan(x) increasing ?.

Best regards.

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up vote 1 down vote accepted

Hint. Draw a picture of $\{z \in \Bbb C : |\mathrm{arg}(z)| \leq \pi/4\}$.

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