Corollary 4.4.37 (Milnor sequence). For every tower \(\dots \to Y_2 \to Y_1 \to Y_0\) of spectra and every \(n \in \Z \), there is a natural short exact sequence \[ 0 \longrightarrow \lim _k^1 \pi _{n+1}(Y_k) \longrightarrow \pi _n\big (\lim _k Y_k\big ) \longrightarrow \lim _k \pi _n(Y_k) \longrightarrow 0. \]
Proof. Apply Lemma 4.4.34 in the opposite stable \(\infty \)-category \(C\catop \). □
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