mathematical induction

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mathematical induction

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mathematical induction (countable and uncountable, plural mathematical inductions)

1. (mathematics) A method of proof which, in terms of a predicate P, could be stated as: if ${\displaystyle P(0)}$ is true and if for any natural number ${\displaystyle n\geq 0}$, ${\displaystyle P(n)}$ implies ${\displaystyle P(n+1)}$, then ${\displaystyle P(n)}$ is true for any natural number n.
   • 2007, Mark Bridger, REAL ANALYSIS: A Constructive Approach, Hoboken, New Jersey: John Wiley & Sons, page 2:

     Mathematical induction is often compared to the behavior of dominos. The dominos are stood up on edge close to each other in a long row. When one is knocked over, it hits the next one (analogous to n in S implies n + 1 in S), which in turn hits the next, etc. If then we hit the first (0 in S), then they will all eventually fall (S is all of ${\displaystyle \mathbb {N} }$). In Variation 1 above, we start by knocking over the kth domino, so that it and all subsequent ones eventually fall.