USAMO 1995 number 5 is a great problem solving example.--MCrawford 15:22, 18 June 2006 (EDT)
"not really a proof"
Any two vectors define a plane; in that plane, we can measure the angle between them, and CS is then equivalent to the fact that the cosine of this angle is less than 1 in absolute value. This is just as much a proof of CS (taking for granted some simple facts about the geometry of ) as it is the reverse -- if you think these facts about are "less basic" than CS, feel free to add a proof of CS using "more basic" things. --JBL 15:58, 9 April 2008 (UTC)
Hm. I agree with you that the cosine way is indeed a valid proof. But both of the proofs of the general form on this page are extremely close to the axioms of fields and inner product spaces, using almost no intermediate results, whereas the axiomatic construction of and derivation of basic trigonometry is pretty nontrivial by comparison. So while I agree with you that it is perfectly valid to go this way, I also think it is valid (and perhaps easier) to construct and define the cosine function analytically, and then to prove the restrictions on its range using one of the general proofs of Cauchy. —Boy Soprano II 18:40, 9 April 2008 (UTC)
No argument from me -- I was commenting on two edits that suggested this wasn't a valid proof. In fact, I think it might be nice to have a more elementary proof of the real case and then a discussion of this question in some appropriately-titled subsection somewhere.
Unrelatedly, why did you get rid of all the \left and \right commands? --JBL 17:07, 10 April 2008 (UTC)
That sounds good to me.
The \left and \right commands generally scale the delimiters to the tallest object inside them. For variable-sized operators with superscripts and subscripts, this typically means they turn out too large. This one of the many issues that the amsmath package was written to address; they talk about it in section 4.14 of the amsmath user's guide, if you want to look there for a better explanation. (It's a good resource for cool LaTeX odds and ends in general; I recommend it highly.) —Boy Soprano II 20:36, 10 April 2008 (UTC)