Exercise 2.2.5

Question

Let $[[x]]$ be the greatest integer less than or equal to $x$. For example, $[[\pi]]=3$ and $[[3]]=3$. For each sequence, find $\lim a_{n}$ and verify it with the definition of convergence.
  \begin{enumerate}[label=(\alph*)] 
  \item $a_{n}=[[5 / n]]$,
  \item $a_{n}=[[(12+4 n) / 3 n]]$.
   \end{enumerate}
  Reflecting on these examples, comment on the statement following Definition 2.2.3 that ``the smaller the $\epsilon$-neighborhood, the larger $N$ may have to be.''

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item For all $n > 5$ we have $[[5/n]] = 0$ meaning $\lim a_n = 0$.
  \item The inside clearly converges to $4/3$ from above, so $\lim a_n = 1$.
   \end{enumerate}
  Some sequences eventually reach their limit, meaning $N$ no longer has to increase.

Exercise 2.2.5

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Exercise 2.2.5

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