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 $$L$$.