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I am currently studying dynamic programming using the Bersketas book: Dynamic programming and optimal control, volume 1. The question is regarding the notation used, but is the following:

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The solutions to certain 'selected' problems are provided online by the author, and this one has me perplexed. In order to prove that this function is monotonically non-decreasing he does the following:

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Now, here is where my confusion stems from: How is the defined function, F, indexed by k, rather than k+1 in the induction proof in both steps? In addition, for the case where i+1=n, why isn't the expression: >= F_k+1 (c(n-1) ... ) ?

Thank you.

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