Formalizing an idea [closed]

Solution 1:

There is a mistake in your argument, which follows from the fact that the complement of this union contains no interval.

Let $C=\bigcup I_n$ the open set which covers the rationals whose measure is at most $\frac19$. You are considering $B=[0,1]\setminus C$, which is a subset of the irrationals.

If $B$ would contain an open interval then it would contain a rational number. Since there are no rational numbers in $B$ there is no interval subset. The irrational numbers form a totally disconnected space, namely every connected component is a singleton.

Note that by DeMorgan we have $B=\bigcap [0,1]\setminus I_n$, this is an intersection of closed sets. Indeed for every $n$ we have that $[0,1]\setminus (I_1\cup\ldots\cup I_n)$ contains an interval, in which there are infinitely many rational and irrational numbers.

The limit, however, is not require to have the properties of the sequence, and we have that $B$ contains no interval.

(This thread can be useful here: Fake induction proof)