Proof by Contradiction Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

hard
Prove by contradiction: There is no largest integer.

Solution

  1. 1
    Assume for contradiction that there exists a largest integer MM.
  2. 2
    Consider M+1M + 1. Since MM is an integer, M+1M + 1 is also an integer, and M+1>MM + 1 > M.
  3. 3
    This contradicts the assumption that MM is the largest integer. Therefore, no largest integer exists.

Answer

ThereĀ isĀ noĀ largestĀ integer.\text{There is no largest integer.}
The proof shows that for any proposed largest integer, we can always construct a larger one, making the assumption impossible.

About Proof by Contradiction

Proof by contradiction (reductio ad absurdum) assumes the negation of what you want to prove, then derives a logical contradiction, thereby establishing that the original statement must be true.

Learn more about Proof by Contradiction ā†’

More Proof by Contradiction Examples

Example 1 medium

Prove by contradiction that [formula] is irrational.

Example 2 medium

Prove by contradiction that there are infinitely many prime numbers.

Example 4 medium

Prove by contradiction that there is no smallest positive rational number.

ā† All Proof by Contradiction Examples Proof by Contradiction Concept