Strike monotonicity and convexity

For European calls on the same underlying and expiry, prove without any model that for $K_1<K_2<K_3$: (a) $c(K_1)\ge c(K_2)$; (b) $c(K_1)-c(K_2)\le e^{-rT}(K_2-K_1)$; (c) $c(K_2)\le\lambda c(K_1)+(1-\lambda)c(K_3)$ where $\lambda=(K_3-K_2)/(K_3-K_1)$. (d) Check (a)–(c) on the Dec settlement prices 24,500 → 587.65, 24,700 → 476.90, 25,000 → 334.95.