Using the Cauchy criterion to disprove a limit

This video demonstrates how to prove that the limit of the sequence $a_n = (-1)^n$ does not exist by using the contrapositive of the Cauchy Convergence Principle. It defines the negative condition ($\exists \varepsilon > 0, \forall N, \dots$), selects $\varepsilon=1$, and uses visual examples with specific indices ($N=3, 6, \dots$) to show that adjacent terms always maintain a distance of at least 2. Finally, it generalizes this finding to conclude that since the sequence is not a Cauchy sequence, its limit cannot exist.

Before you watch

• Definition of Sequence Limits
• Cauchy Sequences
• Real Number Completeness Axiom
• Quantifier Logic (Negation)