Definition:
- We first define three different metrics on $\mathbb R^2$: $(\mathbb{R}^2, d_1)$, $(\mathbb{R}^2, d_2)$ and $(\mathbb{R}^2, d_\infty)$ with
$$d_1(x, y):=|x_1 −y_1|+|x_2 −y_2|$$ $$d_2(x, y) := {\sqrt{(x_1 − y_1)^2 + (x_2 − y_2)^2 }}$$ $$d_\infty(x, y) := \max{|x_1 − y_1|, |x_2 − y_2|}$$ where $x=(x_1,x_2) \in \mathbb{R}^2$ and $y=(y_1,y_2) \in \mathbb{R}^2$.
What we've already proved:
- For all $x,y \in \mathbb{R}^2$, $$d_\infty(x, y) \le d_2(x, y) \le d_1(x, y) ≤ \sqrt{2} d_2(x, y) ≤ 2 d_\infty(x, y)$$
- A sequence $(a_n, b_n)$ converges to $(a, b) \in\mathbb{R}^2$ for the distance $d_\infty(x, y)$ if and only if $a_n$ and $b_n$ respectively converge to $a$ and $b$.
What we're looking for :
- We want to show that $(a_n, b_n)$ converges to $(a, b) \in\mathbb{R}^2$ for the distance $d_\infty(x, y)$ if and only if $(a_n, b_n)$ converges to $(a, b) \in\mathbb{R}^2$ for the distance $d_1(x, y)$, if an only if
$(a_n, b_n)$ converges to $(a, b) \in\mathbb{R}^2$ for the distance $d_2(x, y) $. - We also want to show that open sets for $d_\infty\\$ are open for $d_1\\$ and for $d_2\\$.
Any hints please?
You are dealing with strong equivalence of metrics.
Let $(X,d_1)$ and $(X,d_2)$ be metric spaces and suppose it was given that there exists constants $A,B >0$ such that $$Ad_1(x,y) \leq d_2(x,y) \leq Bd_1(x,y)$$ for all $x,y\in X$. Now, Let $(x_n)_n$ be a sequence in $X$ and $x\in X$.
Claim: $x_n\to_{d_1} x$ $\iff$ $x_n\to_{d_2} x$, where "$\to_{d} x$" means "converges to $x$ with respect to $d$".
Proof: We have that $$0 \leq d_1(x_n,x) \leq \frac{1}{A} d_2(x_n,x)$$ and $$0 \leq d_2(x_n,x) \leq Bd_1(x_n,x).$$ Therefore $$d_1(x_n,x) \to 0 \iff d_2(x_n,x)\to 0,$$ that is $$x_n\to_{d_1} x \iff x_n\to_{d_2} x.$$