(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]X = \textbf{b}[/tex] denote the set of all bounded real-valued

sequences. Define the two metrics:

[tex]

\begin{align*}

d_{\infty}(x,y) := \sup_{n \in \mathbb{N}} |x_n - y_n| \text{,

and } d_e(x,y) := \sum^\infty_{n=1} \frac{1}{2^n} \frac{|x_n -

y_n|}{1 + |x_n - y_n|}

\end{align*}

[/tex]

for [tex]x = (x_1,x_2,\ldots), y = (y_1,y_2,\ldots) \in \textbf{b} = X[/tex].

Prove that these metrics are not equivalent.

2. Relevant equations

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

3. The attempt at a solution

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

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# Homework Help: Equivalent metric functions

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