= Solution
Let $h_t$ and $k_t$ be predictable holdings in the stock and account, with the stochastic integrability needed for a <self-financing portfolio>. Then $V=hS+kB$ and $dV=h\,dS+k\,dB$. The <Itô product rule> and self-financing identity give
$$
\begin{aligned}
d(ZV)&=Z\,dV+V\,dZ+d[Z,V]\\
&=\{ZhS\sigma-ZV\lambda\}\,dW
+\{ZhS\mu+ZkBr-ZVr-ZhS\sigma\lambda\}\,dt.
\end{aligned}
$$
Since $V=hS+kB$ and $\sigma\lambda=\mu-r$, the finite-variation term is zero. Therefore
$$
\boxed{d(ZV)=Z(hS\sigma-V\lambda)dW,}
$$
and the deflated wealth is a <local martingale>.
For <zero-capital nonnegative wealth under a local deflator>, a <nonnegative local martingale> is a <supermartingale>, by localization and the <Conditional Fatou lemma>. Under the required nonnegative-wealth condition, $ZV\ge0$ and starts at zero, so $0\le\mathbb E(Z_tV_t)\le0$. Thus $Z_tV_t=0$ almost surely at every fixed $t$. Strict positivity of $Z$ gives $V_t=0$ almost surely. Taking a countable intersection over rational times and then using continuous wealth paths strengthens this to
$$
\boxed{V_t=0\quad\text{for every }t\ge0\text{ on a single event of probability one}.}
$$
This argument only needs the local deflator, so it remains valid without promoting $ZB$ to a true <martingale>.
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