Limit of a Sequence 01

Calculus Level pending

Let a n a_n be a real-valued sequence such that − 1 < a n < 1 \displaystyle{-1 < a_n < 1} for all n ≥ N n \geq N , for some N ∈ N N \in \mathbb{N} . Which of the following is the STRONGEST true statement that can be made about a n a_n ?

None of these are necessarily true. lim ⁡ n → ∞ a n \lim_{n \rightarrow \infty} a_n exists and − 1 < lim ⁡ n → ∞ a n < 1 -1 < \lim_{n \rightarrow \infty} a_n< 1 lim ⁡ n → ∞ a n \lim_{n \rightarrow \infty} a_n exists and − 1 ≤ lim ⁡ n → ∞ a n ≤ 1 -1 \leq \lim_{n \rightarrow \infty} a_n\leq 1 lim ⁡ n → ∞ a n \lim_{n \rightarrow \infty} a_n exists and is finite.

Daniel Juncos by Daniel Juncos , 45, USA

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1 solution

Daniel Juncos
Sep 17, 2017

Let a n = ( − 1 ) n 2 \displaystyle{a_n = \frac{(-1)^n}{2}} . The a n a_n is 1 2 , − 1 2 , 1 2 , − 1 2 , . . . \displaystyle{\frac{1}{2}, \frac{-1}{2}, \frac{1}{2}, \frac{-1}{2}, ...} ; i.e. − 1 < a n < 1 -1 < a_n < 1 for all n ≥ 0 n \geq 0 , but lim ⁡ n → ∞ a n \displaystyle{\lim_{n \rightarrow \infty} a_n} does not exist.

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