I have an exercise where I need to show that the prices of call options $ C(t,K)=E((S_t-K)^+),t \in [0,T]$ with Strike $K$ for fixed $t$: $$\frac{\partial ^+C(t,K)}{\partial K}=-P(S_t>K).$$ We havent discussed Black Scholes model yet. I guess this will be the introduction exercises for the BS formulas. With: $$\frac{\partial ^+}{\partial K}=\lim_{h↓0}\frac{C(t,K+h)-C(t,K)}{h}$$ I get: $\frac{\partial ^+C(t,K)}{\partial K}=\lim_{h↓0}\frac{C(t,K+h)-C(t,K)}{h}=\lim_{h↓0}\frac{E((S_t-(K+h))^+)-E((S_t-K)^+)}{h}=\lim_{h↓0}\frac{P(S_t>K+h)(E(S_t|S_t>K+h)-(K+h))-P(S_t>K)(E(S_t|S_t>K)-K)}{h}...$
From there I dont know how to proceed further. Using L'Hospital b/c we have $"\frac{0}{0}"$ or left term could be 0. Please help.