Consider the following set $$ I_{n,j}=[\frac{n}{j}-\frac{1}{4^{n+j}},\frac{n}{j}+\frac{1}{4^{n+j}}]$$ for some integers $n$ and $j$. Now let $$A:=\bigcup_{n\geq 1}\bigcup_{j\geq1}I_{n,j}$$
The goal of some exercise I was working on was to show that $A$ is dense in $[0,1]$ and in next step to show that $[0,1]\setminus A\neq \emptyset$.
The first part follows directly as rationals are dense in $\mathbb R$ and I managed to show the second part by using that the Lebesgue-measure of $A$ is strictly smaller then 1.
And so the question appeared if it is possible to show $[0,1]\setminus A\neq \emptyset$ without measure theory and in particular if it is possible to find an explicit element in $[0,1]$ which is not in $A$.
I would appreciate any help.
Thanks in advance!