Let $$\dots \subseteq F_{p - 1}(C) \subseteq F_p(C) \subseteq F_{p + 1}(C) \subseteq \dots$$ be a filtration of a chain complex in an abelian category. In his book Introduction to Homological Algebra, Weibel constructs a spectral sequence associated to this filtration. However, I don't understand a certain step of this construction.
Why do we have $B^r_{p,q} \subseteq Z^s_{p,q}$ for all $r,s \geq 0$?
As Weibel says, this is assembling the definitions, though I don't fault you for getting lost in the notation. As does Weibel, I will omit the $q$:
\begin{align*} B_{p-r}^{r+1}&:=\eta_{p-r}(d(A_{p}^{r}))\\ &=\eta_{p-r}(d(\{c\in F_{p}C\mid d(c)\in F_{p-r}C\}))\\ &=\{\eta_{p-r}d(c)\in E_{p}^{0}\mid c\in F_{p}C,d(c)\in F_{p-r}C\};\\ Z_{p}^{r}&:=\eta_{p}(A_{p}^{r})\\ &=\eta_{p}(\{c\in F_{p}C\mid d(c)\in F_{p-r}C\})\\ &=\{\eta_{p}(c)\in E_{p}^{0}\mid c\in F_{p}C,d(c)\in F_{p-r}C\}. \end{align*} Thus for all $r,s\geq0$, \begin{align*} B_{p}^{r}&=\{\eta_{p}d(c)\mid c\in F_{p+r-1},d(c)\in F_{p}C\}\\ &\subseteq\{\eta_{p}(x)\mid x\in F_{p}C\}\\ &\subseteq\{\eta_{p}(c')\mid c'\in F_{p}C,d(c')\in F_{p-s}C\}\\ &=Z_{p}^{s}, \end{align*} since $d$ need not be surjective to get the first inclusion, and we impose another predicate on the set to get the second inclusion.