Let x y and z be positive real munbers Prove that if xz and

Let x, y, and z be positive real munbers. Prove that if x>z and y^2 = xz, then x > y > z. (by contradiction)

Solution

we know that x>z.

so we have to prove x>y and y>z.

let x<y

as z<x, so z<y.

then x*z<y^2.

but given x*z=y^2.

so our assumption is false and hence x>y.

2) z<x but no comparision between z and y is specified.

now y^2=xz

so y/z=x/y

from previous proof x>y so, x/y>1.

so y/z>1

and hence y>z.

so x>y>z(proved).