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Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.
In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. [15]
Basic arithmetic properties (zoom in for induction proofs) This article contains mathematical proofs for some properties of addition of the natural numbers: the additive identity, commutativity, and associativity. These proofs are used in the article Addition of natural numbers.
In Peano arithmetic, second-order arithmetic and related systems, and indeed in most (not necessarily formal) mathematical treatments of the well-ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic.
The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer n, call the following statement S(n): ( + ) = + . For n > 0, we proceed by mathematical induction.
Pages in category "Mathematical induction" The following 8 pages are in this category, out of 8 total. This list may not reflect recent changes. ...
In mathematics, an elementary proof is a mathematical proof that only uses basic ... small set of axioms concerning integer arithmetic and mathematical induction.
Green's theorem when D is a simple region; Heine–Borel theorem; Intermediate value theorem; Itô's lemma; Kőnig's lemma; Kőnig's theorem (set theory) Kőnig's theorem (graph theory) Lagrange's theorem (group theory) Lagrange's theorem (number theory) Liouville's theorem (complex analysis) Markov's inequality (proof of a generalization) Mean ...