How can the transitivity of the sup metric on bounded sequences be proven?

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In summary, the three parts of the statement are: (1) a sequence is bounded if there is a number M such that |x_{n}| ≤M for all n, (2) d(\{x_{n}\},\{y_{n}\}) is a metric on X, and (3) sup\{|x_{n}-z_{n}| : n \in N \} + sup\{|z_{n}-y_{n}| : n \in N \} is an upper bound for ##\{|x_{n}-y_{n}| :n\in N\}##.
  • #1
bobby2k
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Homework Statement


A sequence [itex]\{x_{n}\}[/itex] of real numbers is called bounded if there is a number M such that |[itex]x_{n}[/itex]| ≤M for all n. Let X be the set of all bounded sequences, show that
[itex]d(\{x_{n}\},\{y_{n}\})=sup \{|x_{n}-y_{n}| :n \in N \}[/itex] is a metric on X.The only part I am struggling with is the transivity part. As I see it, I have to show that:

[itex]sup \{|x_{n}-y_{n}| :n \in N \}≤sup \{|x_{n}-z_{n}| :n \in N \}+sup \{|z_{n}-y_{n}| :n \in N \}[/itex]Do you guys have any tips on how to show this? The problem is that I can not be sure that I have an n where I get the sup value, and even if I did, this n might be different for the three parts.
 
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  • #2
bobby2k said:

Homework Statement


A sequence [itex]\{x_{n}\}[/itex] of real numbers is called bounded if there is a number M such that |[itex]x_{n}[/itex]| ≤M for all n. Let X be the set of all bounded sequences, show that
[itex]d(\{x_{n}\},\{y_{n}\})=sup \{|x_{n}-y_{n}| :n \in N \}[/itex] is a metric on X.


The only part I am struggling with is the transivity part. As I see it, I have to show that:

[itex]sup \{|x_{n}-y_{n}| :n \in N \}≤sup \{|x_{n}-z_{n}| :n \in N \}+sup \{|z_{n}-y_{n}| :n \in N \}[/itex]


Do you guys have any tips on how to show this? The problem is that I can not be sure that I have an n where I get the sup value, and even if I did, this n might be different for the three parts.

Start with ##|x_n-y_n|\le |x_n-z_n|+|z_n-y_n|## and take the sup of both sides. Then compare what that gives with the right side of your expression.
 
  • #3
LCKurtz said:
Start with ##|x_n-y_n|\le |x_n-z_n|+|z_n-y_n|## and take the sup of both sides. Then compare what that gives with the right side of your expression.

It became a little messy, can I simplify?

##
|x_{n}-y_{n}|≤ |x_{n}-z_{n}|+|z_{n}-y_{n}| \forall n ## (1)
##
|x_{n}-z_{n}| ≤ sup\{|x_{n}-z_{n}| : n \in N \} \forall n ## (2)
##|z_{n}-y_{n}| ≤ sup\{|z_{n}-y_{n}| : n \in N \} \forall n ## (3)So if I put 2 and 3 in 1 I get:
##|x_{n}-y_{n}|≤ sup\{|x_{n}-z_{n}| : n \in N \} + sup\{|z_{n}-y_{n}| : n \in N \} \forall n ## (4)

Hence ##sup\{|x_{n}-z_{n}| : n \in N \} + sup\{|z_{n}-y_{n}| : n \in N \} ## is an upper bound for ##\{|x_{n}-y_{n}| :n\in N\}##. So it must also be bigger than the least upper bound for this set. Hence:
##sup\{|x_{n}-y_{n}| : n \in N \} ≤ sup\{|x_{n}-z_{n}| : n \in N \} +sup\{|z_{n}-y_{n}| : n \in N \} ##

Is this the simplest way do it, or is it a simpler way?
 
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  • #4
bobby2k said:
It became a little messy, can I simplify?

##
|x_{n}-y_{n}|≤ |x_{n}-z_{n}|+|z_{n}-y_{n}| \forall n ## (1)
##
|x_{n}-z_{n}| ≤ sup\{|x_{n}-z_{n}| : n \in N \} \forall n ## (2)
##|z_{n}-y_{n}| ≤ sup\{|z_{n}-y_{n}| : n \in N \} \forall n ## (3)


So if I put 2 and 3 in 1 I get:
##|x_{n}-y_{n}|≤ sup\{|x_{n}-z_{n}| : n \in N \} + sup\{|z_{n}-y_{n}| : n \in N \} \forall n ## (4)

Hence ##sup\{|x_{n}-z_{n}| : n \in N \} + sup\{|z_{n}-y_{n}| : n \in N \} ## is an upper bound for ##\{|x_{n}-y_{n}| :n\in N\}##. So it must also be bigger than the least upper bound for this set. Hence:
##sup\{|x_{n}-y_{n}| : n \in N \} ≤ sup\{|x_{n}-z_{n}| : n \in N \} +sup\{|z_{n}-y_{n}| : n \in N \} ##

Is this the simplest way do it, or is it a simpler way?

There may be slightly nicer ways to write it up but I see nothing wrong with your argument.
 

1. What is a metric?

A metric is a quantitative measure of distance or similarity between two objects. It is used to define a mathematical space where calculations can be performed.

2. How do you show that something is a metric?

In order to show that something is a metric, it must satisfy three properties: non-negativity, symmetry, and triangle inequality. Non-negativity means that the distance between two objects must be greater than or equal to zero. Symmetry means that the distance from object A to object B is the same as the distance from object B to object A. Triangle inequality means that the distance from object A to object B is always less than or equal to the sum of the distance from object A to object C and the distance from object C to object B.

3. What are some examples of metrics?

Some examples of metrics include Euclidean distance, Manhattan distance, and Jaccard similarity. Euclidean distance measures the straight-line distance between two points in a plane. Manhattan distance measures the distance between two points by only moving horizontally or vertically. Jaccard similarity measures the similarity between two sets by comparing the number of shared elements to the total number of elements in both sets.

4. Why is it important to show that something is a metric?

Showing that something is a metric is important because it allows for the application of mathematical principles and techniques to solve problems. Metrics are used in various fields such as mathematics, physics, and computer science to analyze and understand data.

5. Can a metric be negative?

No, a metric cannot be negative. As mentioned before, one of the properties of a metric is non-negativity, which means that the distance between two objects must be greater than or equal to zero. If a metric were to have negative values, it would violate this property and would not be considered a metric.

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