Math Problem Statement
Let's tackle this
Solution
Let's break down and solve the given problems one by one:
Problem (b): Metric Space Property
Let be a metric space. The problem asks to show the inequality:
This is an application of the triangle inequality of metric spaces. To prove this:
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Using the triangle inequality: By definition of a metric space, for all , the following holds: Rearranging, we get:
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Switching the roles: Now, switch and :
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Combining both inequalities: We now have the following two inequalities: Thus, the absolute value inequality holds:
This proves the required result.
Problem (c): Function defined by distance
Given the function defined by: where is fixed, we need to show two things:
Part (i): Prove that is uniformly continuous
To show is uniformly continuous, we need to show that for every , there exists a such that for all , if , then .
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Start by expressing :
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Using the result from part (b), we know that:
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Therefore, we have:
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To satisfy , it is enough to choose . That is, if , then:
Thus, is uniformly continuous.
Part (ii): Convergence of a sequence
We are given a sequence in such that: and need to show that converges to .
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From the definition of , the condition means:
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This implies that the distance between and tends to 0, which by the definition of convergence in a metric space means that:
Thus, the sequence converges to .
Do you want further details or have any questions on the steps?
Here are 5 related questions to deepen understanding:
- What is the triangle inequality in a metric space, and why is it essential for proving part (b)?
- How can we generalize the proof of uniform continuity for other types of metric spaces?
- Why does the limit imply that converges to ?
- Can the function be Lipschitz continuous, and how would you show that?
- What is the difference between pointwise and uniform continuity in metric spaces?
Tip: When working with sequences in metric spaces, always think of convergence in terms of distances shrinking to zero between sequence points and the limit.
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Math Problem Analysis
Mathematical Concepts
Metric Space
Uniform Continuity
Sequence Convergence
Triangle Inequality
Formulas
|d(x, z) − d(y, z)| ≤ d(x, y)
f(x) = d(x0, x)
Theorems
Triangle Inequality
Suitable Grade Level
Undergraduate - Advanced Mathematics
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