= Solution
Induct on $n$. The one-dimensional assertion is immediate, and the zero-dimensional case is vacuous. If $T$ has a <cyclic vector> spanning all of $V$, the direct proof in part (ii) applies. Otherwise choose $x\ne0$ and let $W$ be its <cyclic subspace>. Part (i) makes $W$ invariant; the absence of a cyclic vector makes it proper and nonzero.
By induction, the restriction to $W$ is annihilated by $\chi_A$, and the induced map on $V/W$ is annihilated by $\chi_D$. The quotient assertion means $\chi_D(T)V\subset W$. Applying the restriction assertion next gives
$$
\chi_A(T)\chi_D(T)V=0.
$$
By the characteristic factorization in part (iii), this is precisely $\chi_T(T)V=0$. Hence
$$
\boxed{\chi_T(T)=0\quad\text{for every finite-dimensional endomorphism}.}
$$
This is the <Cayley-Hamilton theorem from cyclic subspaces>; using the quotient map is what makes the argument valid even when the block $B$ is nonzero.
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