Elementary proof of Bott periodicity

Question

The elementary proof of Bott periodicity was found by Atiyah and Bott, and published in the paper "On the periodicity theorem for complex vector bundles". In Hatcher's book p.41-51, this elementary proof is called the fundamental product theorem. One thing I don't understand is why, in Hatcher's book, it takes so much to write about convergence issues (even $\epsilon-\delta$ shows up) to show that every clutching function can be approximated by a Laurent polynomial, while in Atiyah and Bott's paper, I don't think any convergence issue is mentioned. I have only read Hatcher's book and I think that step is necessary, so I was wondering why this issue isn't addressed in the Atiyah-Bott paper or how it is bypassed.