How to write induction proof of Sprague-Grundy function for subtration game?

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So lets say that S={1,2,3} I find the sequence of Sprague-Grundy function. How do I justify my answer using induction?

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The proof of the Sprague-Grundy theorem is available at https://en.wikipedia.org/wiki/Sprague%E2%80%93Grundy_theorem#Proof . I'm not sure what a subtraction game is, but if it's an impartial game, then it falls under the Sprague-Grundy theorem.