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$X_t^{x,H}$ is the wealth delivered by the holdings chosen before time $t$, after receiving their time-$t$ dividends. The investor then chooses $H_{t+1}$, whose market value is $H_{t+1}\cdot P_t$, and consumes the remainder $C_t^{x,H}$. A numéraire portfolio is a strategy $\eta$ with zero consumption and strictly positive wealth $N_t=X_t^{\nu ,\eta }$ at every time, where $\nu ={\eta }_1\cdot P_0$.

Write $V_t=X_t^{x,H}-C_t^{x,H}=H_{t+1}\cdot P_t>0$. Set $a_0=1$ and recursivelyThe factors are positive and adapted, so $\eta$ is previsible. For $t\ge 1$,while the same identity at $t=0$ defines $\nu ={\eta }_1\cdot P_0$. Thus consumption is zero and wealth is strictly positive, so $\eta$ is a numéraire portfolio.

A numéraire portfolio $\eta$ satisfies $\eta \cdot P_0>0$ and $\eta \cdot P_1>0$ almost surely. If an arbitrage $H$ already has zero initial cost, it is a terminal-consumption arbitrage. Otherwise $H\cdot P_0<0$; setThen $\tilde{H}\cdot P_0=0$, while its terminal payoff is the nonnegative payoff of $H$ plus a strictly positive multiple of $\eta \cdot P_1$. Hence it is strictly positive almost surely and is a terminal-consumption arbitrage.