$E$ is nowhere dense 
Let $E$ be the set of all $x\in [0.4,0.777\dots]$ whose decimal expansion contains only digits $4$ and $7$. How can we show that $E$ is nowhere dense in $[0.4,0.777\dots]$? That is, there is no interval $(a,b)$ in $E$.

My Try: $E$  does not contain intervals of the form $(0.4\dots41,0.4\dots3)$ and $(0.4\dots48,0.4\dots444)$ etc. But I can't find some 'form' of intervals that are not in $E$, such that every interval that lies in $[0.4,0.777\dots]$ lies in one of those intervals.

Lets say there is interval (a,b). 
Then all numbers between $a$ and $b$ must be in your set. 
Lets take decimal expansion of $a$ and $b$ and find first digit where they differ. Put for example $5$ there. You have a number that is between them, but is not in your set. 
Therefore there can't be any interval.
