 Solution

First, no arbitrage implies the lower boundotherwise buy the call and $K$ maturity-$(T+1)$ bonds and short one non-dividend-paying stock; the initial receipt is positive and the terminal payoff is nonnegative. At time $T$, the assumption $P_T^{T+1}\le 1$ therefore gives $C_T^{T+1,K}\ge (S_T-K{)}^{+}$.

If $C_t^{T,K}>C_t^{T+1,K}$, sell the shorter call and buy the longer one. At time $T$, the longer call's no-arbitrage value covers the shorter call's payoff, with a strictly positive initial receipt. This is impossible, so $T\mapsto C_t^{T,K}$ is nondecreasing.