Prove by contradiction that the harmonic mean of two positive numbers lies between the two numbers

Show that the expression given is an irrational expression

Prove that the square root of two is an irrational number using an indirect proof

Use an indirect proof method to show that the sum of the given trig functions can't be less than one.

Prove a statement about two integers x and y, using an indirect proof method

Prove by contradiction that the following statement about integers x,y, and z cannot be true.

Prove, using an indirect proof, that there is an infinite number of prime numbers

Show that there is no 'smallest' positive rational number using an indirect proof