Lecture 011



Sequence: defined inductively (recursively)

sequence of real numbers

The Factorial Sequence

Conjecture: for all integer n, if n >=4, then n!>2^n Proof: proceed by induction on n\in \mathbb{N} with n \geq 4

The Fibonacci Sequence

Strong Principle of Mathematical Induction


  1. P(0) holds
  2. (\forall k \in \mathbb{N})((\forall i \in [k] \cup \{0\})P(i) \implies P(k+1))

We can use the PMI to prove Strong Principle of Mathematical Induction. Proof: Let P(n) be a variable proposition defined on n \in \mathbb{N} satisfying conditions (1) and (2) of the theorem. Let Q(n) be variable proposition (\forall i \in [n] \cup \{0\}P(i). We proceed by induction on n\in \mathbb{N} to show Q(n) holds for all n.

Show: \implies (\forall n \in \mathbb{N})P(n), fix n. Q(n) \implies P(n) because Q(n) = (\forall i \in [n] \cup {0})P(i) \land n\in[n]\cup {0}.

Template for Strong Induction

Claim: (\forall n \in \mathbb{N})P(n) Proof: We proceed by strong induction on n\in \mathbb{N}

Note: we can use PMI for the number of base case is integer

Claim: \forall n \in \mathbb{N}(f_n < 2^n) Proof:

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