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I am given a set of sequences $S = \{(x_n)_n | x_n \in \{0, 1\}\}$ and the following function: $$d(x, y) = \sum_n \frac{|x_n - y_n|}{2^n}, x, y \in S.$$

Noting that the series is strictly positive, as is the fraction, are the following statements correct?

  1. $d(x,x) = 0 \iff |x_n - x_n| = 0$
  2. $d(x, y) = d(y, x) \iff |x_n - y_n| = |y_n - x_n|$
  3. $d(x, y) + d(y, z) \ge d(x, z) \iff |x_n - y_n| + |y_n - z_n| \ge |x_n - z_n|$
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    They shouldn't be phrased as equivalences (using $\Leftrightarrow$). Instead, the statement $d(x, x) = 0$ is true, with proof: $|x_n - x_n| = 0$ for all $n$. Also, do we assume $n \geq 0$?2011-11-16
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    @Shaun You are right about the first one, but are the next two correctly phrased?2011-11-16
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    To check that $d$ is indeed a distance you also have to show that the series is always convergent. Now, you have to show that the axioms which define a distance are indeed satisfied by $d$. So, you have to check that if $d(x,y)=0$ then the sequences $x$ and $y$ are the same, the symmetry is quite obvious and the triangle inequality follow from what you wrote at 3. after the $\iff$. But you have to write it in a more rigorous way.2011-11-16
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    @PaulManta All of the statements should be phrased in a similar way. What is to the left of $\Leftrightarrow$ is a (true) statement, and you should follow it with the proof of the statement. As to what you have written, your reasoning for 2 and 3 seems to be missing essential details (if this were a HW, I'd grade it less than half credit due to missing details).2011-11-16
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    Hint: The definition of $d(x,y)$ involves a sum and some powers of $2$. Those items ought to be part of your proof as well.2011-11-16
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    See also: http://math.stackexchange.com/questions/472516/how-to-prove-that-this-series-is-a-metric-dx-y-sum-i-0-infty-fracx-i2016-04-01

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I would do minor changes:

  1. $d(x,x) = 0 \iff |x_n - x_n| = 0$; If you used the implication $(\forall n\in\mathbb N)|x_n - x_n| = 0$ $\Rightarrow$ $d(x,x)=0$ instead, it would seem a little clearer to me.

  2. $d(x, y) = d(y, x) \iff |x_n - y_n| = |y_n - x_n|$; Again I would write this as $(\forall n\in\mathbb N) |x_n - y_n| = |y_n - x_n|$ $\Rightarrow$ $d(x, y) = d(y, x)$

  3. $d(x, y) + d(y, z) \ge d(x, z) \iff |x_n - y_n| + |y_n - z_n| \ge |x_n - z_n|$; the same thing here.

In the first two cases it is perhaps not so important, but in the third one the implication $\Longrightarrow$ is not true.


In case you have already learned something about products of topological spaces, you will find out that this is in fact the Cantor space. This metric is explicitly mentioned e.g. in Banach Space Theory: The Basis for Linear and Nonlinear Analysis By Marián Fabian, Petr Habala et al., p.739


BTW is there a name for this metric? It is neither Fréchet metric $d_F$ nor Baire metric $d_B$. $$d_F(x,y)=\sum_{n=1}^\infty\frac1{2^n}\frac{|x_n-y_n|}{1+|x_n-y_n|}$$ $$d_B(x,y)=\frac1{\min\{n\in\mathbb N; x_n\ne y_n\}} \qquad\text{for }x\ne y$$ These two metrics are used frequently for sequences space and their natural generalizations can be used to define topology on countable product of metric spaces.