Question

Let d₁ and d₂ be two metrics for the set X and suppose that there is a positive number c such that d₁(x,y) less than or equal to cd₂ (x,y) for all x,y element of X .Then prove that the identity function , (X,d₁) converges to (X,d₂) is continuous

Answer 1

The identity function ${\rm id}:(X,d_2)\to (X, d_1)$ is a continuous function at a point $x\in X$ if and only if

$\forall \varepsilon>0\exists\delta>0\, \forall x': d_2(x,x')<\delta\to d_1(x,x')<\varepsilon$

For all $\varepsilon>0$ put $\delta=\varepsilon/c$. For every $x'\in X$ if $d_2(x,x')<\delta$ then

$d_1(x,x')\leq cd_2(x,x')<c\delta<\varepsilon$

This means that the identity function from $(X,d_2)$ to $(X,d_1)$ is a continuous function.