SUMMARY
The set of cluster points for the sequence A := {(−1)n/n : n ∈ N} is confirmed to be solely 0. This conclusion is reached by applying the definition of a cluster point, which states that a point x is a cluster point of a set if every neighborhood of x contains at least one point from the set different from x itself. In this case, as n approaches infinity, the terms of the sequence oscillate between positive and negative values, converging to 0, thus establishing it as the only cluster point.
PREREQUISITES
- Understanding of sequences and limits in real analysis
- Familiarity with the definition of cluster points
- Basic knowledge of mathematical proofs
- Experience with oscillating sequences
NEXT STEPS
- Study the definition and properties of cluster points in topology
- Learn about convergence of sequences and their limits
- Explore examples of oscillating sequences and their behaviors
- Practice constructing proofs for identifying cluster points in various sets
USEFUL FOR
Students of real analysis, mathematicians interested in topology, and anyone studying the properties of sequences and convergence.