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Section 5.5 Uniqueness Proofs

Subsection 5.5.1 Uniqueness Proofs

To prove that an object \(A\) is unique, the typical method is to assume there exists two objects \(A\) and \(B\) which both satisfy the desired properties. Then, show that these two objects are equal, \(A = B\text{.}\)

In some sense, this is a proof by contradiction. That is,

  1. By contradiction, assume that there exists \(A, B\) which satisfy the properties, and that \(A \neq B\text{.}\)

  2. Show that \(A = B\text{,}\) contradicting the initial assumption.

Subsection 5.5.2 Examples of Uniquness Proofs

  • The division algorithm

  • Uniqueness of an inverse of a square matrix.

  • Uniqueness of the limit of a function (or sequence).