@Drew Let $L$ be a smashing localization. We show that the acyclics for some telescopic localization are acyclic for $L$. Suppose that $L(K(n)) \ne 0$ and $L(K(n+1)) = 0$. Let $X$ be a finite spectrum of type at least $n+1$ then for $m \le n$ we have $L(X) \otimes K(m) = L(K(m)) \otimes X = K(m) \otimes X = 0$ and for $m \gt n$ we have $L(X) \otimes K(m) = L(K(m)) \otimes X = 0 \otimes X = 0$ so we must have $X=0$.

We now show that the acyclics for $L$ are contained in the acyclics for some $L_n$ for some $n$ (where $L_n$ is $E(n)$-localization. Let $X$ be an $L$ acyclic spectrum. Then $0 …