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1 year ago in Abstract Algebra , Mathematics By Pradeep Kumar

Is it true that for every c > 0, there exists a subsequence satisfying x_{k_n} - x_n ‑ c under certain conditions?

This question arose in my analysis of convergent sequences and cluster points. I'm trying to understand the precise conditions under which one can engineer such a subsequence, as it touches on density properties and the behavior of consecutive terms. A counterexample or constructive proof would be invaluable.

Limit Subsequence Convergence

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By Krupa Answered 1 year ago

This is a subtle problem. The answer is not "always," but it is true under a key condition: if the set of limits of x_{m} - x_{n} (for some suitable pairs) is dense in an interval containing c. I've worked on similar constructions. If, for example, x_{n+1} - x_n → 0 and the sequence is bounded, you can often "steer" the differences. You select indices n and k_n > n such that the difference x_{k_n} - x_n hops around and eventually approximates c arbitrarily closely. It requires a careful iterative argument akin to a bisection method on the index set.

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Is it true that for every c > 0, there exists a subsequence satisfying x_{k_n} - x_n ‑ c under certain conditions?

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