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

Subsequences and limits, proposition

Subsequences and limits

Let \( (a_n)_{n=0}^{\infty} \) be a sequence of real numbers, and let \( L \) be a real number. Then the the following two statements are logically equivalent:

  • The sequence \( (a_n)_{n=0}^{\infty} \) converges to \( L \).
  • [...]

Subsequences related to limit points

Let \( (a_n)_{n=0}^{\infty} \) be a sequence of real numbers, and let \( L \) be a real number. Then the following two statements are logically equivalent:

  • [...]
  • There exists a subsequence of \( (a_n)_{n=0}^{\infty} \) which converges to \( L \).