= Solution
Use the usual normalization $Z_0=1$, finite deterministic initial wealth $x=X_0\geq0$, positive <state-price density>, and nonnegative <consumption>. The deflated asset prices $ZP$ are local <martingales>, so <predictable> stochastic integrability makes $M=\int H\cdot d(ZP)$ a zero-starting <local martingale>. Integrating part (b) gives
$$
N_t:=x+M_t=Z_tX_t+\int_0^tZ_sc_s\,ds\geq0.
$$
A nonnegative <local martingale> is a <supermartingale>: localize to <martingales>, apply <Conditional Fatou lemma>, and obtain the <conditional expectation> inequality and integrability. Therefore $N$, and hence $M=N-x$, are <supermartingales>. In particular,
$$
\boxed{\mathbb E[M_t\mid\mathcal F_s]\leq M_s,\qquad s\leq t.}
$$
This is <supermartingale control of deflated consumption gains>. Nonnegative <consumption> is essential to the lower bound; it is the ordinary meaning of a <consumption> rate here. Wealth nonnegativity alone would not supply that bound if arbitrary signed cash flows were allowed.
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