Exercise 3.3.2

Question

Decide which of the following sets are compact. For those that are not compact, show how Definition 3.3.1 breaks down. In other words, give an example of a sequence contained in the given set that does not possess a subsequence converging to a limit in the set.
  \begin{enumerate}[label=(\alph*)] 
  \item $\mathbf{N}$.
  \item $\mathbf{Q} \cap[0,1]$.
  \item The Cantor set.
  \item $\left\{1+1 / 2^{2}+1 / 3^{2}+\cdots+1 / n^{2}: n \in N\right\}$.
  \item $\{1,1 / 2,2 / 3,3 / 4,4 / 5, \ldots\} .$
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Not compact, the sequence $x_n = n$ in $\mathbf N$ has no convergent subsequence in $\mathbf N$.
  \item Not compact, as we can construct a sequence $(x_n) 	o 1/\sqrt 2 
otin \mathbf Q \cap [0, 1]$ implying $K$ is not closed, and thus cannot be compact.
  \item Compact, since the cantor set is bounded and closed since it is the infinite intersection of closed sets $\bigcap_{n=1}^\infty C_n$ where $C_1 = [0, 1/3] \cup [2/3, 1]$ etc where you keep removing the middle thirds of each interval.
  \item Not compact as every sequence $(x_n)$ contained in the set converges to $\pi^2/6$ which is not in the set, meaning the set isn't closed and thus cannot be compact.
  \item Compact since it is bounded and closed, with every sequence in the set converging to one.
   \end{enumerate}

Exercise 3.3.2

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Exercise 3.3.2

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