Are the metrics d_infty and d_e equivalent?

[complexnumber]

Homework Statement

Let $$X = \textbf{b}$$ denote the set of all bounded real-valued
sequences. Define the two metrics:
$$\begin{align*}<br /> d_{\infty}(x,y) := \sup_{n \in \mathbb{N}} |x_n - y_n| \text{,<br /> and } d_e(x,y) := \sum^\infty_{n=1} \frac{1}{2^n} \frac{|x_n -<br /> y_n|}{1 + |x_n - y_n|}<br /> \end{align*}$$
for $$x = (x_1,x_2,\ldots), y = (y_1,y_2,\ldots) \in \textbf{b} = X$$.

Prove that these metrics are not equivalent.

Homework Equations

For a space $$X \ne \varnothing$$, two distance functions $$d_1,d_2$$ are equivalent if for all sequences $$\{x_k \} \subset X$$ $$\lim_{k \to \infty} d_1(x_k,x) = 0$$ if and only if $$\lim_{k \to \infty} d_2(x_k,x) = 0$$.

The Attempt at a Solution

I guess the proof is to show that for sequence $$\boldsymbol{x}^{(k)}$$, $$\lim_{k \to \infty} d_{e}(x^{(k)}, x) \ne 0$$ when $$\lim_{k \to \infty} d_{\infty}(x^{(k)}, x) = 0$$. But how can I prove this? What area of math do I need?

 

[LCKurtz]

Homework Statement

Let $$X = \textbf{b}$$ denote the set of all bounded real-valued
sequences. Define the two metrics:
$$\begin{align*}<br /> d_{\infty}(x,y) := \sup_{n \in \mathbb{N}} |x_n - y_n| \text{,<br /> and } d_e(x,y) := \sum^\infty_{n=1} \frac{1}{2^n} \frac{|x_n -<br /> y_n|}{1 + |x_n - y_n|}<br /> \end{align*}$$
for $$x = (x_1,x_2,\ldots), y = (y_1,y_2,\ldots) \in \textbf{b} = X$$.

Prove that these metrics are not equivalent.

Homework Equations

For a space $$X \ne \varnothing$$, two distance functions $$d_1,d_2$$ are equivalent if for all sequences $$\{x_k \} \subset X$$ $$\lim_{k \to \infty} d_1(x_k,x) = 0$$ if and only if $$\lim_{k \to \infty} d_2(x_k,x) = 0$$.

The Attempt at a Solution

I guess the proof is to show that for sequence $$\boldsymbol{x}^{(k)}$$, $$\lim_{k \to \infty} d_{e}(x^{(k)}, x) \ne 0$$ when $$\lim_{k \to \infty} d_{\infty}(x^{(k)}, x) = 0$$. But how can I prove this? What area of math do I need?

Think about {x_n} where x_n(k) = δ_nk, the Kronecker delta.

 

[complexnumber]

Think about {x_n} where x_n(k) = δ_nk, the Kronecker delta.

Thank you for the suggestion. I can see this is a very smart counter example. Do you know with this kind of questions if we are supposed to prove by contradiction or a more generic way?

 

[LCKurtz]

I guess the proof is to show that for sequence $$\boldsymbol{x}^{(k)}$$, $$\lim_{k \to \infty} d_{e}(x^{(k)}, x) \ne 0$$ when $$\lim_{k \to \infty} d_{\infty}(x^{(k)}, x) = 0$$. But how can I prove this? What area of math do I need?

Thank you for the suggestion. I can see this is a very smart counter example. Do you know with this kind of questions if we are supposed to prove by contradiction or a more generic way?

You yourself described above what you need to do. Try it with the sequence I suggested.