Uniqueness of limits Prove that limits of sequences are...

Uniqueness of limits Prove that limits of sequences are unique.
That is, show that if $L_{1}$ and $L_{2}$ are numbers such that $a_{n} \rightarrow L_{1}$
and $a_{n} \rightarrow L_{2},$ then $L_{1}=L_{2}$

Key Concepts

Uniqueness of Limits

The uniqueness of limits is a core result in analysis stating that a convergent sequence cannot have more than one limit. This principle underscores the consistency and well-defined nature of limits in real analysis by ensuring that if a sequence converges, then all the arguments based on epsilon closeness must converge to the same value.

Proof by Contradiction

This method involves assuming the opposite of what is to be proven—in this case, that two different limits exist—and then demonstrating that this assumption leads to a logical inconsistency. In proving uniqueness of limits, the contradiction typically arises by showing that the assumed nonzero distance between the two distinct limits violates the convergence conditions imposed by the epsilon definition.

Epsilon Definition of Convergence

This concept defines the limit of a sequence by stating that for every positive number epsilon, no matter how small, there exists a stage in the sequence beyond which all terms differ from the limit by less than epsilon. This rigorous approach ensures that the terms of the sequence eventually get as close as desired to the limiting value.

Triangle Inequality

The triangle inequality is a fundamental property in analysis which says that for any real numbers, the absolute value of a sum is at most the sum of the absolute values. In the context of proving the uniqueness of limits, it is used to relate the distances between the sequence terms and two different candidate limits, helping to establish that these candidate limits must actually coincide.

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