Dyadic Rational are dense I want to prove that set of all dyadic rational numbers in $[0,1]$ is dense in $[0,1]$ but I do not want to prove it using binary expansion of a number. Is there any other proof for the same?

**HINT** $\rm\quad r < \dfrac{m}{2^n} < s\ \iff\ 2^n\: r < m < 2^n\:s\:.\: $ For $\rm\:r < s\:$ such integers $\rm\:m,n\:$ necessarily exist since, by the Archimedean property, $\rm\ 2^n\: (s-r) > 1\ $ for large $\rm\:n\:,\:$ so said interval must contain an integer $\rm\:m\:$ since the interval has length $> 1\:.$