= Solution
Condition on the information available before time $n$. The current wealth is then fixed, while $\xi_n\sim N(b,\Sigma)$ is independent of that information. Put
$$
B=b^T\Sigma^{-1}b>0.
$$
For any portfolio $\theta$, use the <inner product> induced by the <positive-definite matrix> $\Sigma$ to write
$$
\theta=\lambda\Sigma^{-1}b+\eta,
\qquad
\lambda=\frac{\theta^Tb}{B},
\qquad
\eta^Tb=0.
$$
The <linear image of a multivariate normal vector> shows that the two portfolio returns are jointly normal, and
$$
\operatorname{cov}\left((\lambda\Sigma^{-1}b)^T\xi_n,\eta^T\xi_n\right)
=\lambda\eta^T\Sigma\Sigma^{-1}b
=\lambda\eta^Tb=0.
$$
Thus <independence of uncorrelated jointly normal variables> makes the residual return $\eta^T\xi_n$ independent of the aligned return; it also has <expected value> zero.
The continuation value $V(n,\cdot)$ is concave by part (c). Conditional <Jensen inequality> therefore shows that adding the independent centered residual cannot improve the objective:
$$
\mathbb E\left[V\left(n,m+(\lambda\Sigma^{-1}b)^T\xi_n+\eta^T\xi_n\right)\right]
\leq
\mathbb E\left[V\left(n,m+(\lambda\Sigma^{-1}b)^T\xi_n\right)\right],
$$
where $m=(1+r)X_{n-1}$. Since the optimal portfolio is unique, its residual must be zero.
It remains to determine the sign. If $\lambda<0$, the portfolios $\lambda\Sigma^{-1}b$ and $-\lambda\Sigma^{-1}b$ have the same return <variance>, namely $\lambda^2B$, while their means are $\lambda B$ and $-\lambda B$. The latter return has the distribution of the former plus the positive constant $-2\lambda B$. Since $V(n,\cdot)$ is increasing, replacing $\lambda$ by $-\lambda$ cannot reduce the objective, contradicting uniqueness. Hence $\lambda\geq0$.
Applying this conditional argument at every time gives nonnegative, past-measurable random variables $\lambda_n$ such that
$$
\boxed{\theta_n^*=\lambda_n\Sigma^{-1}b},
\qquad 1\leq n\leq N.
$$
This is the <Gaussian one-fund theorem>.
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