Exercise 2.3.12

Question

A typical task in analysis is to decipher whether a property possessed by every term in a convergent sequence is necessarily inherited by the limit. Assume $\left(a_{n}\right) \rightarrow a$, and determine the validity of each claim. Try to produce a counterexample for any that are false.

\begin{enumerate}[label=(\alph*)] 
  \item If every $a_{n}$ is an upper bound for a set $B$, then $a$ is also an upper bound for $B$.
  \item If every $a_{n}$ is in the complement of the interval $(0,1)$, then $a$ is also in the complement of $(0,1)$.
  \item If every $a_{n}$ is rational, then $a$ is rational.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item True, let $s = \sup B$, we know $s \le a_n$ so by the order limit theorem $s \le a$ meaning $a$ is also an upper bound on $B$.
  \item True, since if $a \in (0,1)$ then there would exist an $\epsilon$-neighborhood inside $(0,1)$ that $a_n$ would have to fall in, contradicting the fact that $a_n 
otin (0,1)$.
  \item False, consider the sequence of rational approximations to $\sqrt 2$
   \end{enumerate}

Exercise 2.3.12

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Exercise 2.3.12

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