Nothing makes me stare off into space and wonder why I'm bothering with mathematics at all more than a proof presented with the intent of educating the reader yet the opening statement of which lacks substantive intuitive relevance. So, the proof I want to understand be damned, I come asking for your enlightenment.

The proof is of the Cauchy-Schwarz inequality for complex numbers (its from a complex analysis text, preface says it's intended for a first graduate course). Let

be complex numbers and show that

. The proof begins:

For any complex number

.

Would someone kindly explain how I ought to "know", a priori, to introduce an extraneous complex number, the conjugate, essentially this leading inequality and all it involves, to begin the proof of the CS inequality?