Exercise 2.3.9

Question

\begin{enumerate}[label=(\alph*)] 
  \item Let $\left(a_{n}\right)$ be a bounded (not necessarily convergent) sequence, and assume $\lim b_{n}=0$. Show that $\lim \left(a_{n} b_{n}\right)=0$. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
  \item Can we conclude anything about the convergence of $\left(a_{n} b_{n}\right)$ if we assume that $\left(b_{n}\right)$ converges to some nonzero limit $b$ ?
  \item Use (a) to prove Theorem 2.3.3, part (iii), for the case when $a=0$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item We can't use the ALT since $a_n$ is not necessarily convergent. $a_n$ being bounded gives $|a_n| \le M$ for some $M$ giving
    $$|a_nb_n| \le M|b_n| < \epsilon$$
    Which can be accomplished by letting $|b_n| < \epsilon/M$ since $(b_n) 	o 0$.
  \item No
  \item In (a) we showed $\lim (a_nb_n) = 0 = ab$ for $b = 0$ which proves part (iii) of the ALT.
   \end{enumerate}

Exercise 2.3.9

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Exercise 2.3.9

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