Triangle inequality for complex numbers

This will be a sample formal proof write-up to use as a guide for doingproof write-ups for the class. I went over a proof in the lecture for thismaterial, but a proof during a lecture looks different than a written up proof.Various levels of details can be covered in a proof or omitted depending onthe intended audience. Advanced mathematicians omit many trivial orobvious details since they have done so many proofs over and over again,these details are very apparent to them and they understand how these proofsgo. Since this is a beginning level proof class I will expect to see more detailsshown on the work for proofs.

Theorem 0.1 (triangle inequality for complex numbers). Let z, w C, thefollowing inequality is true:

|z + w| |z|+ |w|.

Proof. By using standard properties of the modulus and conjugate we have

(z + w)(z + w) = |z + w|2

and

(z + w)(z + w) = (z + w)(z + w) = |z|2 + zw + zw + |w|2= |z|2 + zw + zw + |w|2 = |z|2 + 2Re(zw) + |w|2.

Combining these equations and using more basic properties of modulus (in-cluding Re(z) |z|) we have

|z + w|2 = |z|2 + 2Re(zw) + |w|2 |z|2 + 2|zw|+ |w|2= |z|2 + 2|z||w|+ |w|2 = (|z|+ |w|)2.

It then follows that |z + w| |z|+ |w|.

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