I was discussing the proof for the Cauchy-Schwarz inequality used in our lectures, and another student suggested an easier way of doing it. It's really, really simple. But I haven't seen it anywhere online or in textbooks, so I'm wondering if it's either wrong or is only valid in certain scenarios.
The quick version is this:
[itex]u \bullet v = ||u|| ||v|| cosθ[/itex] where θ is between 0 and [itex]\pi[/itex].
The upper and lower bounds of |cosθ| are 1 and 0, respectively, so [itex]|u \bullet v | \leq ||u|| ||v||[/itex]