## Fundamental Principle

Mathematical induction is a method used to prove that a statement holds true for all natural numbers k.

Let P(n) be a statement defined for each positive integer n.

Then P(n) will be true for all positive integers n if the following two conditions are satisfied:

(1) P(1) is true.

(2) P(k) is true for some integer k + 1 implies that P(k + 1) is true.

Because all the other values can use the result from the (1) and (2),

Therefore, you can conclude that all the conditions with integers n are true.

### E1 Example

Q Prove the following for all positive n:

A For n = 1,

The statement is true for n = 1.

For n = k + 1, Assume the statement is true for some integer k ≥ 1.

The statement is true for n = k + 1. By the principle of mathematical induction, the statement is true for all positive integers n.

## Variation

There were cases where proof for each positive integer was needed, instead of all the integers.

### For instance:

When n is in a range.

For all positive odd integers n.

For all positive real numbers.