I'm following the proof of Switzer's "Algebraic Topology and Homotopy" of the known result
Theorem. Let $E$ be a (CW-pre)spectra. There is a natural (up to homotopy) homotopy equivalence $E \wedge \mathbb{S}^1 \to \Sigma E$.
Here $\Sigma E$ is the spectra given by shifting one position, $(\Sigma E)_n = E_{n+1}$.
In the proof given the author defines homotopy inverses over $\mathbb{S}^1 \wedge E_n$ for each $n$ such the natural diagrams commute up to homotopy and pretty much establishes that this finishes the proof (using the generalization of Whitehead's theorem for spectra). What I don't understand is that this seems to use the fact that $\mathbb{S}^1 \wedge E$ is cofinal in $\Sigma E$, but, wouldn't this in turn imply the theorem in a trivial manner? I'm sure there is a subtle problem somewhere but I can't seem to find it. Any light that can be shed on this would be very welcome.
Thanks in advance.