Overview: I'm learning about spectral sequences by working through exercises in Chapter 10 of Bourbaki's Algebra book. One of the exercises makes me suspect that the notion of isomorphism of spectral sequences is poorly behaved in general. Bourbaki seems to think that you can have two spectral sequence which are isomorphic, but which converge to different limits.
(This would be completely outrageous, which is why I'm confused! Hopefully I missed something simple.)
The Problem: Problem 14c of Section 2 asks to show that an isomorphism $u:E\to E'$ of spectral sequences induces isomorphisms $$ B(u_r): B_r \to B_r', \qquad Z(u_r): Z_r \to Z_r', \qquad u_\infty: E_\infty \to E_\infty', $$ of the associated data; for the last two isomorphisms, Bourbaki says we should assume that $E$ and $E'$ are regular (this is a finiteness condition; see the definitions below). Is this assumption strictly necessary? Shouldn't an isomorphism of spectral sequences trivially give isomorphisms of all associated data, including the limits, in all cases?
More concretely,
Is there an isomorphism $u:E\to E'$ of spectral sequences $E$ and $E'$ whose induced map $u_\infty : E_\infty\to E'_\infty$ on the limits is not an isomorphism?
(Bourbaki seems to think there is.)
Definitions: A spectral sequence $E=(E_r,d_r)$ comes with a natural filtration $$ 0 = B_0 \subseteq B_1 \subseteq \cdots \subseteq B_\infty := \bigcup_{r\geq0} B_r \subseteq \bigcap_{r\geq0} Z_r =: Z_\infty \subseteq \cdots Z_1 \subseteq Z_0 = E_0 $$ where $Z_r / B_r$ is isomorphic to $E_r$. We define $$ E_\infty = Z_\infty / B_\infty. $$
The spectral sequence $E$ is regular if for all $(p,q)$ the sequence $(Z_r^{pq})_{r\geq0}$ eventually stabilizes.
A morphism $u:E\to E'$ of spectral sequences $E$ and $E'$ is a sequence of chain maps $(u_r : E_r \to E_r')_{r\geq0}$ such that $u_{r+1}$ is the map on homology induced by $u_r$.
An isomorphism of spectral sequences is a morphism $u:E\to E'$ with an inverse $u':E'\to E$, or equivalently, a morphism $u:E\to E'$ such that for each $r$, the chain map $u_r$ is an isomorphism. (In fact, $u$ is an isomorphism if $u_2$ is.) This definition of isomorphism is my own, not Bourbaki's, but it seems the only natural definition.