# Harmonic Proof

Pro Problems > Math > Logic > Proofs > Indirect Proofs## Harmonic Proof

Create an indirect proof for the following statement: The harmonic mean of any two positive numbers lies inclusively between the two numbers.

## Solution

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.## Similar Problems

### Infinite Primes

Use an indirect proof to show that there are infinitely many prime numbers.

### X Squared Proof

Prove by indirect method that if x and y are integers, then x^{2} - 9y ≠ 3.

### X, Y, and Z proof

Prove by contradiction: for any integers x, y, and z with z > 1, x^{2} - yz^{2} ≠ z.

Hint: This is a more generalized version of the problem found here: X Squared Proof

### Irrational Fraction

Let a and b be positive integers, with b *not *a perfect square. Show that the following expression is irrational, using an indirect proof.

*Hint: Remember that an irrational number is a number which cannot be written as the quotient of two integers.*

### Radical Two Proof

*Hint: An irrational number is a number which cannot be written as the quotient of two relatively prime integers.*

### Sum of Trig Functions

Show that if θ is between 0º and 90º inclusive, then sin θ + cos θ ≥ 1. Use an indirect proof.

### A Smaller Fraction

For every positive rational number x, there exists a smaller positive rational number y. Prove this statement by indirect proof.