Exercise 3.4.6

Question

Prove that $A$ set $E \subseteq \mathbf{R}$ is connected if and only if, for all nonempty disjoint sets $A$ and $B$ satisfying $E=A \cup B$, there always exists a convergent sequence $\left(x_{n}\right) \rightarrow x$ with $\left(x_{n}\right)$ contained in one of $A$ or $B$, and $x$ an element of the other. (Theorem 3.4.6)

Ulisse Mini · Accepted Answer

Both are obvious if you think about the definitions, here's some formal(ish) garbage though

Suppose $\overline{A} \cup B$ is nonempty and let $x$ be an element in both, $x \in B$ implies $x 
otin A$ therefore $x \in L$ (the set of limit points of $A$) meaning there must exist a sequence $(x_n) 	o x$ contained in $A$.

Now suppose there exists an $(x_n) 	o x$ in $A$ with limit in $B$, then clearly $\overline{A} \cap B \subseteq \{x\}$ is nonempty.

Exercise 3.4.6

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Exercise 3.4.6

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