Exercise 4.47 asked you to prove (via algebra) the Arithmetic Mean– Geometric Mean inequality: for x, y ∈ R≥0 , we have √xy ≤ (x + y)/2. Here you’ll reprove the result geometrically. Suppose that x ≥...

Exercise 4.47 asked you to prove (via algebra) the Arithmetic Mean– Geometric Mean inequality: for x, y ∈ R≥0 , we have √xy ≤ (x + y)/2. Here you’ll reprove the result geometrically. Suppose that x ≥ y, and draw two circles of radius x and y tangent to each other, and tangent to a horizontal line. See Figure 4.32(b). Considering the right triangle shown in that diagram, and using the Pythagorean theorem and the fact that the hypotenuse is the longest side of a right triangle, prove the result again.

Exercise 4.47

et ϕ be a fully quantified proposition of predicate logic. Prove that ϕ is logically equivalent to a fully quantified proposition ψ in which all quantifiers are at the outermost level of ψ. In other words, the proposition ψ must be of the form