Sequence bounded away from $0$ and $2$

Question

Suppose I have a sequence of real numbers $\{a_n\}_n$ and I'm told that $\{a_n\}_n$ is bounded away from $0$ and $2$.

(1) What does it mean exactly? My thinking is that it means $a_n\neq 0$ and $a_n \neq 2$ $\forall n$.

(2) Does it imply that $\lim_{n\rightarrow \infty}a_n \neq 0$ and $\lim_{n\rightarrow \infty}a_n \neq 2$ (assuming that the limit exists)?

Question (2) is related to the discussion on rescaling rates at p.211 of van der Vaart "Asymptotic Statistics" point (iii) here where it seems that the answer to (2) is "Yes"

Answer 1

Bounded away from $b$ means that there is a nontrivial interval around $b$ such that the sequence never enters it.

In particular, if a sequence is bounded away from $b$ then it cannot converge to $b$. More generally, $b$ cannot be a limit point of the sequence.

Answer 2

Roughly speaking, it means that the sequence $\{a_n\}$ does not get close to $0$ or to $2$. There exists some $\delta>0$ such that $|a_n-0|=|a_n|>\delta$ for each $n$ and there exists some $\varepsilon>0$ such that $|a_n-2|>\varepsilon$ for each $n$. The limit cannot be $0$ or $2$ if the sequence is bounded away.