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Math and science::Analysis::Tao::06. Limits of sequences

Limit points

Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers, let x be a real number, and let \( \epsilon > 0 \) be a real number. We say that \( x \) is ε-adherent to \( (a_n)_{n=m}^{\infty} \) iff there exists an \( n \ge m \) such that [...]. We say that \( x \) is continually ε-adherent to \( (a_n)_{n=m}^{\infty} \) if it is ε-adherent to [...] for every \( N \ge m \). We say that \( x \) is a limit point or adherent point of \( (a_n)_{n=m}^{\infty} \) if it is continually ε-adherent to [...] for every [...]

How are limit points different to limits?