about the intersection of nested intervals Consider a sequence $\{a_n\}$ (we have not informations about its convergence) and moreover consider a sequence of semi-open intervals of $\mathbb R$:
$$\left[\frac{a_0}{2^0},\frac{a_0+1}{2^0}\right[\supset \left[\frac{a_1}{2^1},\frac{a_1+1}{2^1}\right[\supset\cdots\supset\left[\frac{a_n}{2^n},\frac{a_n+1}{2^n}\right[\supset\cdots$$
Can I conclude that the intersection is only a point? Be aware of the fact that I can't use the Cantor intersection theorem since my intervals are not closed!
Many thanks in advance.

As Daniel Fischer noted, the length of the intervals shrinks to $0$ so the intersection is either empty or contains exactly one point.
If you choose $a_n = 0$ for each $n$, the intervals will be nested and their intersection is $\{0\}$. On the other hand, if you choose $a_n$ so that $\{\frac{a_n + 1}{2^n}\}$ is constant, i.e. $a_{n + 1} = 2a_n + 1$, and $a_0 ≥ 0$ then the intervals will be again nested but the intersection will be empty.
