I'm struggling with the proof of the LOP. The task is the following: There are two self financing strategies $\psi$ and $\theta$ in a multi period Market $(S^0,S^1,...,S^d)$ and $V_T^\psi$ = $V_T^\theta$ holds $P$-a.s.
a) If the Market is arbitrage free, then $V_t^\psi$ = $V_t^\theta$ holds $P$-a.s.
b) If the Market is arbitrage free with $V_T^\psi = S_T^i$ $i=0,1,...d$ then $V_t^\psi = S_t^i$ holds $P$-a.s.
My idea is to proof by contradiction. So I said that there is a t where $V_t^\psi$ = $V_t^\theta$ doesn't hold, so lets assume that $V_t^\psi$ > $V_t^\theta$ holds. Now I want to get a strategy $\psi-\theta$ that is an arbitrage strategy. My teacher told me to have this strategy as the following: The strategy is $0$ from $0$ to $t$. At $t+1$ I get the strategy $\psi-\theta$. But now I dont know how to show that this is arbitrage free. Do I have to do it with the stochastic integral?
Thanks in advance!