Sets $A, B \sube X$ are separated if $A \cap \bar{B} = \bar{A} \cap B = \empty$.
Heuristically, we can say two sets are separated if there isn’t a smooth transition between them; i.e. they do not share limit points.
A set $E \sube X$ is disconnected if it can be written as the union of two separated sets. Moreover, $E$ is said to be connected if it is not disconnected.
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Connectedness Characterization in $\R$.
$$ E\sube \R \text{ is connected} \iff \forall x , y \in E , x<z<y \implies z \in E $$
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A sequence in a set $A$ is a function $f : \N \rightarrow A$ denoted $\{x_n\}$ s.t. $f(n) = x_n$.
A sequence $\{x_n\}$ in a metric space $X$ converges if there exists a point $x \in X$ such that the terms in the sequence get arbitrarily close to $x$ as the index $n$ increases.
$$
\lim _{n\rightarrow \infty} \{x_n\} = x \iff \forall \varepsilon \in \R _{>0}, \exists N \in \N \mid \forall n \geq N, d(x_n, x) <\varepsilon
$$
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$\{n^2\} \sub \R$ diverges.
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Convergence Open Neighborhood Characterization. A sequence $\{x_n\}$ in a metric space $X$ converges to $x \in X$ iff every open neighborhood of $x$ contains $x_n$ for all but finitely many $n$.
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Limits are Unique. If $\{x_n\}$ converges to both $x, x' \in X$, then $x= x'$.
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Convergence requires Boundedness. If $\{x_n\}$ converges, then its range $f(\N)$ is bounded.