Let S be a non-empty convex set in $mathbb{R}^n$ and $f:S rightarrow mathbb{R}^n$. Then f is convex if and only if for each integer $k>0$

$x_1,x_2,…x_k in S, displaystylesumlimits_{i=1}^k lambda_i=1, lambda_igeq 0, forall i=1,2,s,k$, we have $fleft ( displaystylesumlimits_{i=1}^k lambda_ix_i right )leq displaystylesumlimits_{i=1}^k lambda _ifleft ( x right )$

Proof

By induction on k.

$k=1:x_1 in S$ Therefore $fleft ( lambda_1 x_1right ) leq lambda_i fleft (x_1right )$ because $lambda_i=1$.

$k=2:lambda_1+lambda_2=1$ and $x_1, x_2 in S$

Therefore, $lambda_1x_1+lambda_2x_2 in S$

Hence by definition, $fleft ( lambda_1 x_1 +lambda_2 x_2 right )leq lambda _1fleft ( x_1 right )+lambda _2fleft ( x_2 right )$

Let the statement is true for $n

Therefore,

$fleft ( lambda_1 x_1+ lambda_2 x_2+….+lambda_k x_kright )leq lambda_1 fleft (x_1 right )+lambda_2 fleft (x_2 right )+…+lambda_k fleft (x_k right )$

$k=n+1:$ Let $x_1, x_2,….x_n,x_{n+1} in S$ and $displaystylesumlimits_{i=1}^{n+1}=1$

Therefore $mu_1x_1+mu_2x_2+…….+mu_nx_n+mu_{n+1} x_{n+1} in S$

thus,$fleft (mu_1x_1+mu_2x_2+…+mu_nx_n+mu_{n+1} x_{n+1} right )$

$=fleft ( left ( mu_1+mu_2+…+mu_n right)frac{mu_1x_1+mu_2x_2+…+mu_nx_n}{mu_1+mu_2+mu_3}+mu_{n+1}x_{n+1} right)$

$=fleft ( mu_y+mu_{n+1}x_{n+1} right )$ where $mu=mu_1+mu_2+…+mu_n$ and

$y=frac{mu_1x_1+mu_2x_2+…+mu_nx_n}{mu_1+mu_2+…+mu_n}$ and also $mu_1+mu_{n+1}=1,y in S$

$Rightarrow fleft ( mu_1x_1+mu_2x_2+…+mu_nx_n+mu_{n+1}x_{n+1}right ) leq mu fleft ( y right )+mu_{n+1} fleft ( x_{n+1} right )$

$Rightarrow fleft ( mu_1x_1+mu_2x_2+…+mu_nx_n+mu_{n+1}x_{n+1}right ) leq$

$left ( mu_1+mu_2+…+mu_n right )fleft ( frac{mu_1x_1+mu_2x_2+…+mu_nx_n}{mu_1+mu_2+…+mu_n} right )+mu_{n+1}fleft ( x_{n+1} right )$

$Rightarrow fleft ( mu_1x_1+mu_2x_2+…+mu_nx_n +mu_{n+1}x_{n+1}right )leq left ( mu_1+ mu_2+ …+mu_n right )$

$left [ frac{mu_1}{mu_1+ mu_2+ …+mu_n}fleft ( x_1 right )+…+frac{mu_n}{mu_1+ mu_2+ …+mu_n}fleft ( x_n right ) right ]+mu_{n+1}fleft ( x_{n+1} right )$

$Rightarrow fleft ( mu_1x_1+mu_2x_2+…+mu_nx_n+mu_{n+1}x_{n+1}right )leq mu_1fleft ( x_1 right )+mu_2fleft ( x_2 right )+….$

Hence Proved.