The text gives the following definition: "a set $\{I_n\mid n \in \mathbb N\}$ of nonempty closed bounded intervals is called nested intervals (clumsy, I know, I am translating from the German here), iff $ \text {(a) }\forall n \in \mathbb{N}: \quad I_{n+1} \subseteq I_n$ and$\text{(b) }\forall \varepsilon>0 \exists N \in \mathbb{N}: \quad\left|I_N\right|<\varepsilon$." The exercise is then to prove that there exists a (unique) $x\in \mathbb R$ such that $ x = \bigcap_{n \in \mathbb N}I_n$. The proof of uniqueness was easy enough, but I am far from confident of my proof of existence, which goes as follows:
Assume for a contradiction that no such $x\in \mathbb R$ exists, so $\bigcap_{n\in \mathbb N} I_n= \emptyset$. (Here comes the implication I am not sure about) It then follows that there exists an $n\in \mathbb N$ such that $I_{n+1}\nsubseteq I_n$, because were this not the case, then by the definition of (non empty) subsets all $I_n$ would share at least one element. This is a contradiction to (a).