Consider a family of call options on a non$-$dividend$-$paying stock, each option being identical except for its strike price. Suppose the interest rate is $0.$ The value of the call with strike price $K$ is denoted by $C(K).$ Prove the following two general relations using arbitrage arguments:

$($a$)K_2>K_1$ implies $K_2-K_{1}C(K_1)-C(K_2).$

$($b$)K_3>K_2>K_1$ implies

$C(K_2)(\frac{K_3-K_2}{K_3-K_1})C(K_1)+(\frac{K_2-K_1}{K_3-K_1})C(K_3).$