Prove: let x and y be positive real numbers. if x ≤ y, then √x ≤ √y.

Suppose that and that . We can subtract from both sides to obtain . Recall that . Now we can replace and in this identity with and respectively and we get . From this it follows that . Since , we can divide by it both sides of the inequality without altering its direction and end up with . Now we just need to add to both sides and conclude that , which finishes our proof.