For every x∈Rn and every δ>0 define the ``rectangle'' R(x,δ):=(x1−δ,x1+δ)×(x2−δ,x2+δ)×⋯×(xn−δ,xn+δ). Show that these sets generate the same open sets as the balls in standard metric. That is, show that a set U⊂Rn is open in the sense of the standard metric if and only if for every point x∈U, there exists a δ>0 such that R(x,δ)⊂U.