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## Homework Statement

$$

\left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right |

$$

Where z and w are complex numbers not equal to zero.

2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$

## The Attempt at a Solution

EDIT: INCORRECT WORK[/B]

Alright, the following was my approach. I first tried to split up the problem due to symmetry.

$$

\left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right | \left | z \right | \leq \left | z+w \right |

$$

$$

\left | \frac{1}{\bar{z}} + \frac{1}{\bar{w}} \right | \left | z \right | \leq \left | z+w \right |

$$

$$

\left | \frac{\bar{z}+\bar{w}}{\bar{z}\bar{w}} \right | \left | z \right | \leq \left | z+w \right |

$$

And I was left with the following

$$

\left | \bar{z}+\bar{w} \right | \left | z \right | \leq \left | z+w \right | |

\bar{z}\bar{w}|

$$

I feel like I'm either close, or I've worked the problem entirely the wrong way including conjugates.

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