mathematical induction

1. Overview

Definition: Mathematical induction is a proof technique used to establish the truth of an infinite number of statements, typically involving integers.
Two Main Components:
- Base Case: Show that the statement holds for the initial value (usually n=1).
- Inductive Step: Assume the statement is true for some arbitrary integer k (inductive hypothesis) and then prove it must also be true for k+1.
Structure of Induction:
1. Base Case (n=1): Verify the assertion is correct for the first integer.
2. Inductive Hypothesis: Assume the assertion holds for n=k.
3. Inductive Step: Prove the assertion holds for n=k+1 using the inductive hypothesis.
Recursive Structures: Induction is closely related to recursion in computer science, where a problem is solved by reducing it to smaller instances of the same problem.