Proving Numerical Equivalence of Real Number Intervals with S-B Theorem

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The discussion centers on using the Schroeder-Bernstein Theorem to demonstrate that any two intervals of real numbers are numerically equivalent. The theorem states that if there are injections from set A to set B and from B to A, a bijective correspondence exists between them. The original poster expresses difficulty in starting the proof and seeks guidance on how to proceed. They suggest visualizing the intervals on a graph to conceptualize the injective and bijective functions. The conversation emphasizes the importance of establishing the necessary injections to apply the theorem effectively.
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Homework Statement



Using the Schroeder-Bernstein Theorem, prove that any two intervals of real numbers are numerically equivalent.

Homework Equations



Schroeder-Bernstein Theorem: Let A and B be sets, and suppose that there are injections from A into B and B into A. Then, there exists a bijective correspondence between A and B.

The Attempt at a Solution


None. I'm stuck. Can anyone help me with where to go?
 
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If and only if two intervals are "numerically equivalent", there exists a bijective correspondence between A and B.
 
Picture one of the intervals in the x-axis of the plane and the other in the y-axis. Can't you see the graph of an injective (in fact bijective) function between them?
 

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