Uniqueness of Limits

analysis calculus

Definition

Uniqueness of Limits

If a sequence or function has a limit, then that limit is unique. That is, if and , then .

Proof

Uniqueness of Limits

Suppose for the sake of contradiction that a function has two distinct limits and as , where .

1. Let . Since , we have .
2. By the definition of a limit, there exists such that for all :
3. Similarly, there exists such that for all :
4. Let . For any satisfying , both inequalities hold. Using the triangle inequality:
5. Substituting the bounds:

This results in the contradiction , proving that must equal .

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