I have observed that IV is increasing with time to maturity by using market prices and plotting IV (from Black-Scholes) against log-moneyness, $\log(S_t/K)$. $S_t$ being the price of the stock at time $t$ and $K$ being the strike.
Using Martingales we can prove that the call-option's payoff function - i.e. $\max(S_t-K, 0)$ - is a submartingale under the $Q$-measure. Now this article from Columbia says that the call-price as a function of time to expiry, that is $C_t(T)$, must be not-decreasing to avoid arbitrage, which can be shown using standard martingale results - but why is that?
What are the calculations performed by "standard martinale results" which imply that if the call price was decreasing as a function of $T$ then there would be an arbitrage?