Exercise 1.4.6

Question

Recall that a set $B$ is dense in $\mathbf{R}$ if an element of $B$ can be found between any two real numbers $a<b$. Which of the following sets are dense in $\mathbf{R}$ ? Take $p \in \mathbf{Z}$ and $q \in \mathbf{N}$ in every case.
  \begin{enumerate}[label=(\alph*)] 
  \item The set of all rational numbers $p / q$ with $q \leq 10$.
  \item The set of all rational numbers $p / q$ with $q$ a power of 2 .
  \item The set of all rational numbers $p / q$ with $10|p| \geq q$.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Not dense since we cannot make $|p|/q$ smaller then $1/10$.
  \item Dense.
  \item Not dense since we cannot make $|p|/q$ smaller then $1/10$.
   \end{enumerate}

Exercise 1.4.6

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Exercise 1.4.6

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