# Proving exactness in homotopy exact sequence

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 $j_*[g] = [f].$

As you can see in the picture below, I tacked the homotopy, which I named $$H$$, to $$f$$ which is possible since $$H_0 = f$$. Also, $$\partial D^n$$ is now completely mapped to $$x_0$$, which makes the constructed map $$g$$ one of the form $$(D^n, \partial D^n, s_0) \to (X,x_0,x_0)$$, as desired. $g$">
To get $$j_*[g] = [f]$$, you push in/out the cylinder via a homotopy $$G$$ as in the remaining three pictures below.