Assumption 2.1 - If the payoff $P$ of a financial instrument is non negative, then the price $p$ of the financial instrument is non negative.
Assume $C$ is just the price of the call option, and $C^{*}$ is also the price of a different call option with the same parameters as $C$ except that $\tau = T - t$ where $T$ is the maturity and $t$ is current time
Assume no dominance, Assumption 2.1. Show that the price of call option should satisfy $$(S - B_t(T)K)_+ \leq C(T,K,S)\leq C^{*}(T,K,S)\leq S$$ Therefore, for any price quote $C^{*}(\tau,K,S)$ of a call option with strike $K$ and time to maturity $\tau$, there exists a unique $\sigma^{imp}(\tau,K,S)$ such that $$C(\tau,K,S,\sigma^{imp},r) = C^{*}(\tau,K,S)$$ $\sigma^{imp}(\tau,K,S)$ is called implied volatility. See figure below
Attempted proof: Let the price of a call option with strike $K$ be denoted $C(K)$. Assume when we purchase a call option the stock price $S$ is equal to the strike price $K$. Hence we have $$C(T,K,S)\leq S$$ at time $T$.
I am not really sure where to go from here, any suggestions is greatly appreciated.