I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the statement of said theorem in any other source, so I am completely at a loss how to proceed.
The result is called the Eilenberg-Moore Filtration Sequence for complete complexes
Suppose the filtered chain complex $C$ is complete with respect to it's filtration $F_*C$, i.e $C=\lim\limits_{\leftarrow} C/F_pC.$ In this case the exact sequence $$0\longrightarrow\mathbin{\lim\limits_{\longleftarrow}}^1H_{n+1}(C/F_pC)\longrightarrow H_n(C)\longrightarrow\mathbin{\lim\limits_{\longleftarrow}} H_n(C/F_pC)\longrightarrow0$$ takes the form $$0\longrightarrow\bigcap F_pH_n(C)\longrightarrow H_n(C)\longrightarrow H_n(C)/\bigcap F_pH_n(C)\longrightarrow0$$ and $${\lim\limits_{\longleftarrow}} H_n(C/F_pC)\cong {\lim\limits_{\longleftarrow}} H_n(C)/F_pH_n(C). $$
The first exact sequence follows from the general properties of the $\mathbin{\lim\limits_{\longleftarrow}}^1$ functor (theorem 3.5.8):
Let $C_i$ be a tower of chain complexes satisfying the Mittag-Leffler condition, and set $C=\mathbin{\lim\limits_{\longleftarrow}} C_i$. Then there is an exact sequence for each n: $$0\longrightarrow\mathbin{\lim\limits_{\longleftarrow}}^1H_{n+1}(C_i)\longrightarrow H_n(C)\longrightarrow\mathbin{\lim\limits_{\longleftarrow}} H_n(C_i)\longrightarrow0$$
The proof then proceeds as follows:
Taking the limit of the exact sequence of towers $$0\longrightarrow F_pH_*(C)\longrightarrow H_*(C)\longrightarrow H_*(C)/F_pH_*(C)\longrightarrow0$$ we find that there is a monomorphism $H_*(C)/\bigcap F_pH_*(C)\to {\lim\limits_{\longleftarrow}} H_*(C)/F_pH_*(C)$. Also, there is an exact sequnce $$0\longrightarrow H_*(C)/F_pH_*(C)\to H_*(C/F_pC),$$ which also gives a monomorphism ${\lim\limits_{\longleftarrow}} H_*(C)/F_pH_*(C)\to \lim\limits_{\longleftarrow} H_*(C/F_pC)$. This combines into a monomorphism $H_*(C)/\bigcap F_pH_*(C)\to\lim\limits_{\longleftarrow} H_*(C/F_pC)$.
Then Weibel says to "combine this with the $\mathbin{\lim\limits_{\longleftarrow}}^1$ sequence of 3.5.8". And I don't get this last step at all. It seems we have not yet connected $\mathbin{\lim\limits_{\longleftarrow}}^1$ and $\bigcap F_pH_n(C)$.
This result is later used to prove the Complete Convergence Theorem, so it is kind of important. I would be thankful for any help here.
The monomorphism $H_*(C)/\bigcap_p F_pH_*(C) \to \mathbin{\lim\limits_{\longleftarrow}} H_*(C/F_p C)$ is actually an isomorphism, because it is induced by the natural map $\pi: H_* (C) \to \mathbin{\lim\limits_{\longleftarrow}} H_*(C/F_p C)$, which is surjective by $3.5.8.$
This implies that $\bigcap_p F_pH_*(C)$ is $\ker(\pi)$, which is isomorphic to $\mathbin{\lim\limits_{\longleftarrow}}^1 H_{*+1}(C/F_p C)$ by $3.5.8.$