Show discrete market is arbitrage free if and only if there exist no admissible arbitrage portfolios

Question

Problem: Let S be a discrete market. Show S is arbitrage free if and only if there exist no admissible arbitrage portfolios.

1. Definition of Discrete Market: Let $T$ be a positive real number and $N$ be a positive integer. Let $0=t_0<t_1<\cdots<t_N=T$. Let $(\Omega, \mathscr{F}, P)$ be a probability space, where $\Omega$ is finite, $\mathscr{F}=2^{\Omega}$, and $P\{\omega\}>0$ for every $\omega \in \Omega$. A discrete market is a $(\boldsymbol{d}+1)$-dimensional stochastic process $$ S=\left\{\left(S_n^{(0)}, S_n^{(1)}, \ldots, S_n^{(d)}\right): n=0,1, \ldots, N\right\} $$ defined on the probability space $(\Omega, \mathscr{F}, P)$. The first entry $S_n^{(0)}$ is the price at time $t_n$ of a riskless asset and is deterministic, that is, $$ S_0^{(0)}=1, \quad S_n^{(0)}=S_{n-1}^{(0)}\left(1+r_n\right), \quad n=1,2, \ldots, N, $$ where $r_n>-1$ is the risk-free interest rate in the $n$th period $\left[t_{n-1}, t_n\right]$.

   For $j \geq 1$, the $j$ th entry $S_n^{(j)}$ is the price at time $t_n$ of the $j$ th risky asset and follows the following stochastic dynamics: $$ S_0^{(j)}>0, \quad S_n^{(j)}=S_{n-1}^{(j)}\left(1+\mu_n^{(j)}\right), \quad n=1,2, \ldots, N $$ Here $S_0^{(j)}$ is a real number and $\mu_n^{(j)}>-1$ is a real random variable. The amount of information available in the market at time $t_n$ is represented by the $\sigma$-algebra $\mathscr{F}_n$ given by $$ \mathscr{F}_0=\{\emptyset, \Omega\} $$ and $$ \mathscr{F}_n=\sigma\left(\left(\mu_i^{(1)}, \mu_i^{(2)}, \ldots, \mu_i^{(d)}\right): i=1,2, \ldots, n\right), \quad n=1,2, \ldots, N . $$ We assume that $\mathscr{F}_N=\mathscr{F}$.

2. Definition of Arbitrage: Let $\mathcal{A}$ be the family of all self-financing, predictable portfolios of the discrete market. A portfolio $\alpha \in \mathcal{A}$ is called an arbitrage if $V_0(\alpha)=$ 0 and there exists $n \geq 1$ such that $P\left\{V_n(\alpha) \geq 0\right\}=1$ and $P\left\{V_n(\alpha)>0\right\}>0$. The discrete market is called arbitrage-free if $\mathcal{A}$ does not contain arbitrage portfolios.

3. Definition of Admissible Portfolio: A portfolio $\alpha$ is called admissible if for every $n$ in $\{1,2, \ldots, N\}$, we have $$ P\left\{V_n(\alpha) \geq 0\right\}=1 $$

4. Definition of Portfolio: A portfolio or strategy is a $(d+1)$-dimensional stochastic process $$ \alpha=\left\{\left(\alpha_n^{(0)}, \alpha_n^{(1)}, \ldots, \alpha_n^{(d)}\right): n=0,1, \ldots, N\right\} . $$ The value of the portfolio $\alpha$ at time $t_n$ is $$ V_n(\alpha)=\sum_{j=0}^d \alpha_n^{(j)} S_n^{(j)} $$

5. Self financing portfolio definition: A portfolio $\alpha$ is self-financing if for every n in $\{1,2,...,N\}$, we have $V_{n-1}(\alpha)=\sum_{j=0}^d \alpha_n^{(j)} S_{n-1}^{(j)}$

Question: For the if part of the problem, I was told to use method of contradiction, i.e. find a admissible arbitrage portfolio and lead to contradiction. Based on definitions, I haven't figured out how to construct such admissible arbitrage portfolio. I checked https://math.stackexchange.com/questions/2998811/show-that-the-market-is-arbitrage-free-if-a-s-011r-b and have not figured out how to apply it here. Thanks in advance.

Some attempt: I am trying to prove by contradiction that if S is arbitrage free, then there is no admissible portfolio. I am thinking about defining portfolio value $\alpha$ as following piecewise function:

$V_n=0$ when discrete market S is defined on $\Omega \setminus F$, where $F \subset \Omega$

$V_n=0$ when discrete market S is defined on $F$ when $n<N$ for $n=\{1,2,...,N\}$

$V_n=1$ when discrete market S is defined on $F$ when $n=N$. This way it is an admissible arbitrage portfolio but it violates assumption that S is arbitrage free, i.e. there exists no $n \geq 1$ so that $P(V_n(\alpha) \geq 0\}=1$ and $P\{V_n(\alpha)>0\}>0$.Not sure if this approach is general or not.