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**cribby** 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 $\displaystyle z_j,\,w_j$ be complex numbers and show that $\displaystyle \,\,|\sum_1^n z_j w_j |^2 \leq\,\, \sum_1^n |z_j|^2 \sum_1^n|w_j|^2$. The proof begins:

For any complex number $\displaystyle \lambda,\, 0 \leq \sum_1^n|z_j - \lambda \bar{w_j}|^2$.

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?