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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 $$L$$.
• 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.

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