Consider a family of call options on a non-dividend-paying stock, each option being identical except for its strike price. The value of the call with strike price \(K\) is denoted by \(C(K)\). Prove the following three general relations using arbitrage arguments:

(a) \(K_{2}>K_{1}\) implies \(C\left(K_{1}\right) \geq C\left(K_{2}\right)\).

(b) \(K_{2}>K_{1}\) implies \(K_{2}-K_{1} \geq C\left(K_{1}\right)-C\left(K_{2}\right)\).

(c) \(K_{3}>K_{2}>K_{1}\) implies \[
C\left(K_{2}\right) \leq\left(\frac{K_{3}-K_{2}}{K_{3}-K_{1}}\right) C\left(K_{1}\right)+\left(\frac{K_{2}-K_{1}}{K_{3}-K_{1}}\right) C\left(K_{3}\right) .
\]