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**slevvio** Let X be a discrete random variable and $\displaystyle g: R_X \rightarrow \mathbb{R} $ a continuous function, which is convex, i.e. for all $\displaystyle x_1,x_2 \in R_X $ and $\displaystyle \lambda \in (0,1) $

$\displaystyle g(\lambda x_1 + (1-\lambda)x_2 \leq \lambda g(x_1) + (1-\lambda) g(x_2). $

(a) For $\displaystyle \lambda_i \geq 0 $ with $\displaystyle \sum_{i=1}^{n} \lambda_i = 1 $ show that $\displaystyle g(\sum_{i=1}^{n} \lambda_i x_i ) \leq \sum_{i=1}^{n} \lambda_i g(x_i) $

I have let P(n) denote the above statement and shown that it holds for n = 1, but I am finding it hard to progress much further. I really need help with this and any hints or tips would be appreciated.