Suppose that we have a directed system $(E_N^{pq}, f_{N,N'})_{N,N' \in \mathbb{N}}$ of spectral sequences, and that, moreover, for any $N$, the spectral sequence $E_N^{**}$ collapses at its $E_2$-page $$E_{2,N}^{**} = E_{\infty,N}^{**}.$$

The direct limit $E^{**} := \underset{N \to \infty}{\lim} E_N^{**}$ is a spectral sequence, and moreover, it collapses at its $E_2$-page, with $E_2^{**} = \underset{N}{\lim} E_{\infty,N}^{**}.$

I’m wondering if the following holds: suppose that for any $N$, the spectral sequence $E_N^{**}$ converges to $H_N^*$, and that we have maps $H_N^* \to H_{N'}^*$ compatible with the directed system above. Is it true that $E^{**}$ converges to the limit $\underset{N \to \infty}{\lim} H_N^*$ ?

Thanks a lot


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