7-13

(i)\phi_1 and \phi_2 are convex in (a,b), for a<x_1<x_2
\phi_1(\theta x_1+(1-\theta)x_2)\le \theta \phi_1(x_1)+(1-\theta)\phi_1 x_2 and \phi_2(\theta x_1+(1-\theta)x_2)\le \theta \phi_2(x_1)+(1-\theta)\phi_2 x_2,
(\phi_1+\phi_2)(\theta x_1+(1-\theta)x_2)=\phi_1(\theta x_1+(1-\theta)x_2)+\phi_2(\theta x_1+(1-\theta)x_2)
\le \theta \phi_1(x_1)+(1-\theta)\phi_1 x_2+\theta \phi_2(x_1)+(1-\theta)\phi_2 x_2=\theta (\phi_1+\phi_2)(x_1)+(1-\theta)(\phi_1+\phi_2) x_2
Hence,
(\phi_1+\phi_2)(\theta x_1+(1-\theta)x_2)\le \theta (\phi_1+\phi_2)(x_1)+(1-\theta)(\phi_1+\phi_2) x_2
\phi_1+\phi_2 is convex in (a,b).
(ii)\phi is convex in (a,b),for a<x_1<x_2
\phi(\theta x_1+(1-\theta)x_2)\le \theta \phi(x_1)+(1-\theta)\phi x_2,
for every positive constant C,
(c\phi)(\theta x_1+(1-\theta)x_2)=c(\phi)(\theta x_1+(1-\theta)x_2)
\le c(\theta \phi(x_1)+(1-\theta)\phi x_2)=\theta (c\phi)(x_1)+(1-\theta)(c\phi) x_2),
(c\phi)(\theta x_1+(1-\theta)x_2)
\le \theta (c\phi)(x_1)+(1-\theta)(c\phi) x_2)
c\phi is convex in (a,b).
(iii)\phi_k \rightarrow \phi, for all k=1,2,...,a<x_1<x_2
\phi_k(\theta x_1+(1-\theta)x_2)\le \theta \phi_k(x_1)+(1-\theta)\phi_k x_2,
lim_{k\rightarrow\infty}\phi_k(\theta x_1+(1-\theta)x_2)\le \theta lim_{k\rightarrow\infty} \phi_k(x_1)+(1-\theta)lim_{k\rightarrow\infty}\phi_k x_2,
\phi(\theta x_1+(1-\theta)x_2)\le \theta \phi(x_1)+(1-\theta)\phi x_2,
\phi is convex in (a,b).