In first place, I'm using the definition of spectral sequence given by Weibel at the beginning of section 5.2, in AITHA. Hence, in an abelian category $\sf A$, a spectral sequence consists of an object $E^r_{pq}$ for every three integers $p,q,r$ with $r\ge a$, where $a$ is a fixed; and obviously, there are some more axioms that the family $\{E_{pq}^r\}$ has to satisfy, to be a spectral sequence.
I don't understand the definition of convergence of a spectral sequence:
What confuse me is how we define $s,t$; they depend only on $n$, so I just see the following way to make sense. Fixed $n$, let $s'$ be the smallest integer such that $E^a_{sq}$ is nonzero, for some $q$ with $s'+q=n$, and let $t'$ be the largest integer such that $E^a_{tq}$ is nonzero, for some $q$ with $t'+q=n$. Then choose the $s,t$ of Weibel so that $s\lt s'$ and $t\gt t'$.
In this way for any $E^a_{pq}$, letting $s,t$ be the integers relative to $p+q$, we are sure that $s\lt p\lt t$. However I see no trace of this condition in the book: maybe it is implicit that $s,t$ must be chosen in a way that makes sense? But even in this (unlikely I think) case, is this the only way making sense? It seems more probable that I'm making confusion with conditions not needed, but I really can't solve my doubts. Any help is appreciated, thank you.
In fact, we don't care about what the actual values of $s,t$ are; we just want to know that they exist, for technical reasons.
So it would be perhaps clearer to consider, for every $n$, an increasing filtration $(F_kH_n)_{k \in \mathbb{Z}}$ on $H_n$ which is finite in the following sense: $F_kH_n$ is $0$ when $k$ is small enough and $H_n$ when $k$ is large enough.