Given a sequence $x_n$ whose number of limit points is finite. How to algebraically construct a polynomial whose roots are the limit points of a given sequence? Moreover, without explicitly calculating the limit points of this sequence. What ideas can I use here?
2026-04-07 01:55:35.1775526935
How to construct a polynomial?
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Since you are dealing with arbitrary sequences, that is not possible. An arbitrary sequence may have infinitely many limit points and then, unless every real number is a limit point of the sequence, there is no polynomial whose set of roots is the set of the limit points of the sequence.