I will use this theorem 3.2 from the book "Quantitative modeling of Derivative Securities" by Marco Avellandea:
Theorem 3.2 - Assume that there is no arbitrage, i.e. there exists a risk neutral probability $\pi$. Then, the market is complete if and only if there is a unique risk-neutral probability, i.e. the system of linear equation $$p = D\pi$$ has a unique solution where $D$ is are the different future scenarios in the market.
Show that in any model if there are two distinct risk-neutral probabilities, then there are infinite number of them.
Attempted solution: Consider a one-period binomial model:
State 1: $S_0 u$ with risk-neutral probability $\pi_u$
State 2: $S_0 l$ with risk-neutral probability $\pi_l$
Assume $\pi_u,\pi_l$ are two distinct risk-neutral probabilities, i.e., $\pi_u + \pi_l \neq 1$. Then, this violates theorem 3.2, the market is not complete since there cannot exist a unique risk-neutral probability, i.e. the system of linear equations $$p = D\pi$$ does not have a unique solution. Thus if there is not a unique solution then there are an infinite number under our example.
I am not sure if this suffices, any suggestions is greatly appreciated.