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.