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I have a question on Rudin's proof of Theorem 1.33 part e. Here he prove the following statement:

The absolute value of z+w is equal or smaller than the absolute value of z plus the absolute value of w -Yes, is the triangle inequality, where z and w are both complex numbers-

|z+w| $\leqslant$ |z| + |w|

In the proof, the key is that he points out that

$$2Re(z\overline{w}) \leqslant 2|z\overline{w}|$$

which obviously implies that

$$Re(z\overline{w}) \leqslant |z\overline{w}|$$

Why is that so? How does he knows this inequality is satified? If you can help me I would appreciate it a lot.

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# Baby Rudin Proof of Theorem 1.33 (e) - Triangle Inequality

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