The notion of proof is central to mathematics, providing a rigorous way to establish the truth of mathematical statements. One commonly used method of proof is mathematical induction. Let’s discuss both:

1. Introduction to Proof:
   • In mathematics, a proof is a logical argument that demonstrates the validity of a mathematical statement.
   • A proof typically consists of a series of logical steps, starting from known or assumed facts (axioms or previously proven theorems) and using deductive reasoning to derive the desired conclusion.
   • The goal of a proof is to establish that a statement (conjecture or theorem) holds true under all possible conditions or within a specified domain.
2. Mathematical Induction:
   • Mathematical induction is a powerful technique used to prove statements about natural numbers.
   • It consists of two steps: a. Base Case: Proving that the statement holds true for some initial value, typically the smallest value of the domain (often
     $$n=0 or n=1). b. Inductive Step
     

      

     k (the induction hypothesis), and then proving that it also holds true for the next value
     $\mathrm{<br>}$

     k+1.

   • If both the base case and the inductive step are valid, mathematical induction guarantees that the statement holds true for all natural numbers greater than or equal to the initial value.
   • Mathematical induction is often used to prove statements about sums, products, divisibility, inequalities, and other properties of natural numbers.
   • There are variations of mathematical induction, such as strong induction and complete induction, which involve different assumptions or methods for proving the inductive step.

Mathematical induction is a fundamental proof technique in mathematics, widely used in various branches such as number theory, combinatorics, algebra, and analysis. It provides a systematic and reliable method for establishing the truth of statements involving natural numbers.