\(
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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 \( L \).
