Exercise 2.2.7

Question

Here are two useful definitions:
  \begin{enumerate}[label=(\roman*)] 
  \item A sequence $\left(a_{n}\right)$ is \emph{eventually} in a set $A \subseteq \mathbf{R}$ if there exists an $N \in \mathbf{N}$ such that $a_{n} \in A$ for all $n \geq N$.
  \item A sequence $\left(a_{n}\right)$ is \emph{frequently} in a set $A \subseteq \mathbf{R}$ if, for every $N \in \mathbf{N}$, there exists an $n \geq N$ such that $a_{n} \in A$.
    \begin{enumerate}[label=(\alph*)] 
    \item Is the sequence $(-1)^{n}$ eventually or frequently in the set $\{1\}$?
    \item Which definition is stronger? Does frequently imply eventually or does eventually imply frequently?
    \item Give an alternate rephrasing of Definition 2.2.3B using either frequently or eventually. Which is the term we want?
    \item Suppose an infinite number of terms of a sequence $\left(x_{n}\right)$ are equal to 2 . Is $\left(x_{n}\right)$ necessarily eventually in the interval $(1.9,2.1) ?$ Is it frequently in $(1.9,2.1) ?$
     \end{enumerate}
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Frequently, but not eventually.
  \item Eventually is stronger, it implies frequently.
  \item $(x_n) 	o x$ \emph{if and only if} $x_n$ is eventually in any $\epsilon$-neighborhood around $x$.
  \item $(x_n)$ is frequently in $(1.9, 2.1)$ but not necessarily eventually (consider $x_n = 2(-1)^n$).
   \end{enumerate}

Exercise 2.2.7

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Exercise 2.2.7

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