Question

Prove algebraically that |w+z| is less than or equal to |w|+|z| for any complex numbers w and z, where || is the magnitude.

After letting w = a+bi and z = c + di and doing some plugging in etc. I got that

ac-bd <= b^2 + sqrt (a^2+b^2)(c^2+d^2) + d^2

But is my proof complete? I know logically sqrt (a^2+b^2)(c^2+d^2) is a bit less than ac+ad+bc+bd but now what?

Answer 1

There are lots of proofs of the triangle inequality. One such can be found here:

https://www.math.ucdavis.edu/~forehand/67_summer2012_files/triangle%20inequality.pdf