An illuminating construction in Real Analysis

I posted a question on . The answer is brilliant and very illuminating.

The statement of the problem is:
Let us consider $\mathbb{Q}\cap[0,1]$ , putting all the elements in the set in a sequence, denoted $\{a_n \}$ . We define $[a_j-\frac{1}{2^{i+j}} , a_j +\frac{1}{2^{i+j}}]\cap[0,1]$ . Notice that $\mathbb{Q}\subset V_i$ .

So we define $V=\bigcap_i V_i$ . We have $\mathbb{Q}\subset V$ , and $V$ is a zero-measure set.

So how to find an irrational number in $V$ ?

Robert Israel gives a very clever solution:
For an “explicit” construction, consider $z=\sum_{i=1}^{\infty}2^{-i_k}$ , where given $k_1 <\dots<k_n$ , if $\sum_{i-1}^n2^{-k_i}=a_{m(n)} , k_{ n+1} =k_n +1+2m(n)$ . Note that $a_{m(n)} <z<a_{m(n)} +2^{ 1−k_{n+1}} <a_{m(n)} +2^{−2m(n)} so $z\in V_{m(n)}$ . Since $m(n)\to \infty \mbox{ as } n\to \infty$ , $z$ is in all the $V_i$ . Its base-2 expansion contains infinitely many 1’s but also has arbitrarily large gaps between the 1’s, so can’t be eventually periodic, and thus $z$ must be irrational.

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Marine's Math