induction (e) Consider the following solitaire game: Take a piece of paper, and

Question

induction

(e) Consider the following solitaire game: Take a piece of paper, and write as many distinct positive integers on it as you want. A move in this game consists of the following: -Pick any positive integer k on the page. -Erase it. -Write as many new positive integers less than k on the page as you want^2. The only rule is that you cannot have any repeated positive integers on your page after you do this. Prove that no matter what you do. this game eventually ends: that is. that you cannot play forever.

Explanation / Answer

Given the description of the problem, and since we have infinite number of integers and conceptually we are allowing the possibility of being able to write as many integers on the paper as one wishes, erasing only one integer at any point of time and adding as many less than that number does not mean that the gave ever ends. The game goes for ever.

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