having a bit of trouble with this exercise;
Let f be a function, convex on $\mathbb{I}$, by writing $\sum_{k=1}^{n}x_k\lambda_k $ for $ n \ge 3 $ under the form $x_n\lambda_n+(1-\lambda_n)y_n$ with $y_n$ expressed with $x_k, k\in [|1,n|] $ and $ \lambda_q, q\in[|1,n|]$ and $\lambda_i\in]0,1[$and $\sum_{n}^{i=1}\lambda_i = 1$ . Prove that : $\sum_{k=1}^{n}\lambda_kf(x_k)\ge f(\sum_{k=1}^{n}x_k\lambda_k)$.
The previous question was to demonstrate that ; $z\in ]x,y[ <=> \exists\lambda, z=x\lambda + (1-\lambda)y$
hint
$$\sum_{i=1}^n\lambda_ix_i=$$
$$\lambda_nx_n+(1-\lambda_n)\sum_{i=1}^{n-1}\frac{\lambda_i x_i}{1-\lambda_n}=$$
$$\lambda_nx_n+(1-\lambda_n)y_n$$