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 \( L \).