 Solution

For $K_1<K_2$, the lower-strike payoff dominates:If $C_t^{T,K_1}<C_t^{T,K_2}$, buy the cheaper lower-strike call and sell the higher-strike call. This gives positive initial consumption and a nonnegative terminal payoff, an arbitrage. Hence the monotonicity of a European call price in strike gives $C_t^{T,K_1}\ge C_t^{T,K_2}$.