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 \( L \).
- Every subsequence of \( (a_n)_{n=0}^{\infty} \) 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:

- \( L \) is a limit point of \( (a_n)_{n=0}^{\infty} \)
- There exists a subsequence of \( (a_n)_{n=0}^{\infty} \) which converges to \( L \).

These two propositions show a sharp contrast between limits and limit points.