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  2. Mathematical induction - Wikipedia

    en.wikipedia.org/wiki/Mathematical_induction

    The most common form of proof by mathematical induction requires proving in the induction step that (() (+)) whereupon the induction principle "automates" n applications of this step in getting from P(0) to P(n). This could be called "predecessor induction" because each step proves something about a number from something about that number's ...

  3. Proofs involving the addition of natural numbers - Wikipedia

    en.wikipedia.org/wiki/Proofs_involving_the...

    We prove commutativity (a + b = b + a) by applying induction on the natural number b. First we prove the base cases b = 0 and b = S(0) = 1 (i.e. we prove that 0 and 1 commute with everything). The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a.

  4. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    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 ]

  5. Proof by exhaustion - Wikipedia

    en.wikipedia.org/wiki/Proof_by_exhaustion

    Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. [1]

  6. Bernoulli's inequality - Wikipedia

    en.wikipedia.org/wiki/Bernoulli's_inequality

    Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form: we prove the inequality for r ∈ { 0 , 1 } {\displaystyle r\in \{0,1\}} ,

  7. AM–GM inequality - Wikipedia

    en.wikipedia.org/wiki/AM–GM_inequality

    For the following proof we apply mathematical induction and only well-known rules of arithmetic. Induction basis: For n = 1 the statement is true with equality. Induction hypothesis: Suppose that the AM–GM statement holds for all choices of n non-negative real numbers. Induction step: Consider n + 1 non-negative real numbers x 1, . . . , x n+1, .

  8. List of mathematical proofs - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_proofs

    Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational

  9. Proofs of Fermat's little theorem - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_Fermat's_little...

    This proof, due to Euler, [3] uses induction to prove the theorem for all integers a ≥ 0. The base step, that 0 p ≡ 0 (mod p), is trivial. Next, we must show that if the theorem is true for a = k, then it is also true for a = k + 1. For this inductive step, we need the following lemma. Lemma.