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Principle of Mathematical Induction (PMI) Let p (1) , p (2) , ..., p ( k ) , ... be sentences having the following two properties: 1) p (1) is true; 2) ∀ k ∈ N [ p ( k ) is true ⇒ p ( k + 1) is true ] . Then ∀ n ∈ N p ( n ) is true. Example:
Let's prove the Bernoulli inequality: ∀ n ∈ N 1 + nx 6 (1 + x ) n , x − 1. The initial step: For n = 1 we have 1 + x 6 1 + x - true. The inductive step: We assume that for k ( k 1) there is:
1 + kx 6 (1 + x ) k , x − 1. Under this assumption we have to
prove the inequality: 1 + ( k + 1) x 6 (1 + x )( k +1), x − 1.
1 + kx 6 (1 + x ) k | (1 + x ) ⇒ (1 + kx )(1 + x ) 6 (1 + x ) k +1
1 + ( k + 1) x + kx 2 6 (1 + x ) k +1.
Since kx 2 is positive, we have:
1 + ( k + 1) x 6 1 + ( k + 1) x + kx 2 6 (1 + x ) k +1 ⇒ 1 + ( k + 1) x 6 (1 + x )( k +1) . Both assumptions of PMI are
satisfied, therefore the conclusion is true.
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