Question

show that the inequalities satisfies for all point x,y∈R

d*(x,y) ≤d(x,y)≤√n d*(x,y)

Answer 1

Given inequalities is-

$d^*(x,y)\le d(x,y)\le \sqrt{n}d^*(x,y)$

As x and y belongs to the R i.e. $x,y\in R$

$d^*(x,y)\le d(x,y)~~~~-(1)$

According the rolle's theorem -

$d(x,y)\le \sqrt{n}d^*(x,y)~~~~~~-(2)$

From eqn.(1) and eqn.(2) we have-

$d^*(x,y)\le d(x,y)\le \sqrt{n}d^*(x,y)$