K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results.  For instance, here's a scattering of proofs of the Bott periodicity theorem for topological complex K-theory that I've found in the literature:

* Bott's original proof used Morse theory, which reappeared in Milnor's book _Morse Theory_ in a much less condensed form.
* Pressley and Segal managed to produce the homotopy inverse of the usual Bott map as a corollary in their book _Loop Groups_.
* Behrens recently produced a novel proof based on Aguilar and Prieto, which shows that various relevant maps are quasifibrations, therefore inducing the right maps on homotopy and resulting in Bott periodicity.
* Snaith showed that $BU$ is homotopy equivalent to $CP^\infty$ once you adjoin an invertible element.  (He and Gepner also recently showed that this works in the motivic setting too, though this other proof relies on the reader having already seen Bott periodicity for motivic complex K-theory.)
* Atiyah, Bott, and Shapiro in their seminal paper titled _Clifford Modules_ produced an algebraic proof of the periodicity theorem.
* Atiyah and Bott produced a proof using elementary methods, which boils down to thinking hard about matrix arithmetic and clutching functions.  Variations on this have been reproduced in lots of books, e.g., Switzer's _Algebraic Topology: Homotopy and Homology_.
* A proof of the periodicity theorem also appears in Atiyah's book _K-Theory_, which makes use of some basic facts about Fredholm operators.  A differently flavored proof that also rests on Fredholm operators appears in Atiyah's paper _Algebraic topology and operations on Hilbert space_.
* Atiyah wrote a paper titled _Bott Periodicity and the Index of Elliptic Operators_ that uses his index theorem; this one is particularly nice, since it additionally specifies a fairly minimal set of conditions for a map to be the inverse of the Bott map.
* Seminaire Cartan in the winter of '59-'60 produced a proof of the periodicity theorem using "only standard techniques from homotopy theory," which I haven't looked into too deeply, but I know it's around.

Now, for my question: the proofs of the periodicity theorem that make use of index theory are in some vague sense appealing to the existence of various Thom isomorphisms.  It seems reasonable to expect that one could produce a proof of Bott periodicity that explicitly makes use of the facts that:

1. The Thom space of the tautological line bundle over $CP^n$ is homeomorphic to $CP^{n+1}$.
2. Taking a colimit, the Thom space of the tautological line bundle over $CP^\infty$ is homeomorphic to $CP^\infty$.
3. The difference of the tautological line bundle and the trivial bundle over $CP^\infty$ gives a stable equivalence between that Thom space and $\Sigma^{-2} CP^\infty$, and in fact this is the Bott map.

But I can't find anything about this in the literature.  Any ideas on how to squeeze a proof out of this -- or, better yet, any ideas about where I can find someone who's already done the squeezing?

Hope this isn't nonsense!

-- edit --

Given the positive response but utter lack of answers, I thought I ought to broaden the question a bit to start discussion.  What I was originally looking for was a moral proof of the periodicity theorem -- something short that I could show to someone with a little knowledge of stable homotopy as why we should expect the whole thing to be true.  The proofs labeled as elementary contained too much matrix algebra to fit into parlor talk, while the proofs with Fredholm operators didn't seem -- uh -- homotopy-y enough.  While this business with Thom spaces over $CP^\infty$ seemed like a good place to look, I knew it probably wasn't the only place.  In light of Lawson's response, now I'm sure it isn't the only place!

So: does anyone have a good Bott periodicity punchline, aimed at a homotopy theorist?

(Note: I'll probably reserve the accepted answer flag for something addressing the original question.)