Consider the stable homotopy category $\mathcal{SHC}$, we then can define its full subcategory $\mathcal{SHC}^c$ spanned by finite spectra which coincides with the smallest thick triangulated subcategory of $\mathcal{SHC}$ generated by $\mathbb{S}$.
I was told that $\mathcal{SHC}^c$ admits a filtration given by the following sets
$C_{(0)}= \{ \Sigma^l \mathbb{S} \ : \ l \in \mathbb{Z} \}$ $C_{(n+1)}= \{ Z : \textit{there exists a distinguished triangle} \ X \rightarrow Y \rightarrow Z \rightarrow \Sigma X \ \textit{with} \ X, Y \in C_{(n)} \}$
and the category of $p$-local compact spectra $\mathcal{SHC}^c_p$ inherits a similar filtration since it's the $p$-localization of $\mathcal{SHC}^c$.
I tried to prove that $K= \bigcup_{n \in \mathbb{N}} C_{(n)}$ coincides with $\mathcal{SHC}^c$: the inclusion $K \subset \mathcal{SHC}^c$ is trivial, for the other I tried to prove that $K$ is a thick triangulated subcategory of $\mathcal{SHC}$ but I cannot show $K$ is thick.
If someone has any idea how to prove the thickness of $K$ he is welcomed but I suspect that the proof needs some machinery of the stable homotopy category and not only basic theory of triangulated categories.
This is a known fact but I could not find a precise reference in the literature: if you can provide one I will consider it a satisfactory answer.
Thanks in advance for any help.