Showing a call price is convex in strike

Prove that the price of a European call, viewed as a function of its strike, is convex.

The payoff is convex in $K$, and pricing is a positive linear operation. For $\lambda\in[0,1]$, $(S-\lambda K_1-(1-\lambda)K_2)^+\le\lambda(S-K_1)^++(1-\lambda)(S-K_2)^+$ pathwise, so by no-arbitrage $C(\lambda K_1+(1-\lambda)K_2)\le\lambda C(K_1)+(1-\lambda)C(K_2)$. Equivalently, a butterfly spread has a non-negative payoff and so a non-negative price.