Exercise 2.2.4

Question

Give an example of each or state that the request is impossible. For any that are impossible, give a compelling argument for why that is the case.
  \begin{enumerate}[label=(\alph*)] 
  \item A sequence with an infinite number of ones that does not converge to one.
  \item A sequence with an infinite number of ones that converges to a limit not equal to one.
  \item A divergent sequence such that for every $n \in \mathbf{N}$ it is possible to find $n$ consecutive ones somewhere in the sequence.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item $a_n = (-1)^n$
  \item Impossible, if $\lim a_n = a 
e 1$ then for any $n \ge N$ we can find a $n$ with $a_n = 1$ meaning $\epsilon < |1 - a|$ is impossible.
  \item $a_n = (1, 2, 1, 1, 3, 1, 1, 1, \dots)$
   \end{enumerate}

Exercise 2.2.4

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Exercise 2.2.4

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