Summary
Problem: Prove that for vectors $ v, w$ in an inner product space, the inequality $$\displaystyle |\left \langle v, w \right \rangle | \leq \| v \| \| w \|$$ Solution: There is an elementary proof of the Cauchy-Schwarz inequality (see the Wikipedia article), and this proof is essentially the same. What makes this proof stand out is its insightful technique, which I first read about on Terry Tao’s blog. He calls it “textbook,” and maybe it is for an analyst, but it’s still very elegant....