I am trying to work through Hatcher's proof (page 344 of his book) that the homotopy sequence of a triple $$... \rightarrow \pi_n(A,B,x_0) \stackrel{i_*}{\rightarrow} \pi_n(X,B,x_0) \stackrel{j_*}{\rightarrow} \pi_n(X,A,x_0) \stackrel{\partial}{\rightarrow} \pi_{n-1}(A,B,x_0) \rightarrow ..$$ is exact. Specifically, I want to show that $\mathrm{Ker}(\partial) \subseteq \mathrm{Im}(j_*).$
If $f$ represents a class $[f] \in \mathrm{Ker}(\partial)$, then we choose a homotopy $F$ between $f|_{I^{n-1}}$ and a map with image in $B$. Hatcher says "we can tack $F$ onto $f$ to get a new map ... which... is homotopic to $f$ by the homotopy that tacks on increasingly longer initial segments of $F$". I don't understand this, and I haven't been able to write down any map $g$ with $j_*[g] = [f].$