Let $\overline{S}=\{S^0,S^1,...S^n\}$ be a one-period market model with one riskless asset and $n$ risky assets, where $S_t^i$ denotes the value of asset $i$ at time $t$. Further assume that $\overline{S}$ is nonredundant, i.e. for a portfolio $\overline{\theta} \in \mathbb{R} ^{n+1}$, $\overline{\theta} \cdot\overline{S}=0 $ $\implies$ $\overline{\theta}=0$. Denote $X^i$ as the discounted value of asset $i$, that is, $X_t^i=\frac{S_t^i}{S_t^0}$. Finally, denote $\Delta X_1$ as the discounted gains, i.e. $\Delta X_1 = X_1-X_0$. Prove the following $$\theta \cdot \Delta X_1=0 \implies \theta =0$$
For those of you unfamiliar with the notation, the portfolio vector $\overline{\theta}$ denotes the number of shares of asset $i$ to purchase. Note that $\overline{\theta} \in \mathbb{R}^{n+1}$ but $\theta\in\mathbb{R}^n$ due to the discounting.
I've tried a few methods but nothing has worked. I tried to express $\Delta X_1^i$ as a linear combination of $\Delta X_1^j$'s but could not figure out how to proceed. Any insight would be helpful.
In some textbooks the riskless asset price is $S^{0}= 1+r$, with $r \geq 0$ and constant, so let´s assume that $S^{0} \not = 0$.
Suppose that $\theta \in \mathbb{R}^{d}$ is such that $\theta \cdot \Delta X_{1} = 0$ then $\theta \cdot X_{1} = \theta \cdot X_{0}$, if $\bar{\theta} = (-\frac{1}{S^{0}}\theta \cdot X_{0} , \theta_{1},...,\theta_{d})$ therefore:
$$\bar{\theta} \cdot \bar{S} = \bar{\theta_{0}}(S^{0}) + \theta \cdot X_{1} = -\theta \cdot X_{0} + \theta \cdot X_{1} = 0 $$
By non-redundancy $\bar{\theta} = 0 $ and in particular $\theta = 0$.