Definition of the limit of a sequence

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SUMMARY

The limit of a sequence is defined mathematically as lim n → ∞ un = l if for every ε > 0, there exists an N such that for all n > N, |un - l| < ε. The discussion highlights the equivalence of using |un - l| < kε, where k is a positive constant, emphasizing that this does not alter the fundamental definition. Participants argue for clarity in analysis texts regarding the use of kε, suggesting that it can simplify proofs without losing rigor. The conversation also touches on the logical implications of limits and the importance of understanding ε and N in mathematical proofs.

PREREQUISITES
  • Understanding of limits in sequences
  • Familiarity with ε-δ definitions in calculus
  • Basic knowledge of mathematical proofs
  • Concept of convergence in sequences
NEXT STEPS
  • Study the ε-δ definition of limits in depth
  • Explore the concept of convergence and divergence in sequences
  • Learn about the implications of using kε in proofs
  • Investigate the logical structure of mathematical proofs, particularly in analysis
USEFUL FOR

Students preparing for university-level mathematics, particularly those studying analysis, as well as educators seeking to clarify the concept of limits in sequences.

  • #31
Definitions written in symbols:

Given a sequence xn, we say that x is a cluster point for xn if (\forall \epsilon &gt;0)(\forall N)(\exists n&gt;N)(\vert x-x_{n} \vert &lt;\epsilon).

Given a sequence xn, we say that xn converges to x if (\forall \epsilon &gt;0)(\exists N)(\forall n&gt;N)(\vert x-x_{n} \vert &lt;\epsilon).
 

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