---- > [!definition] Definition. ([[sequence]]) > A **sequence** $(x_{n})_{n \in \mathbb{N}}$ is a function with domain $\mathbb{N}$. It is thought of as an enumerated collection of objects in which repetitions are allowed and order matters. > [!definition] Definition. (Metric-Specific Convergence). > A sequence $(v_{n})_{n \in\mathbb{N}}$ of points in a [[metric space]] $X$ is said to **converge** to $v \in X$ provided that for all $\varepsilon > 0$, there exists $N \in \mathbb{N}$ s.t. for all $n > N$, the [[open ball]] of radius $\varepsilon$ centered at $v$, $B_{\varepsilon}(v)$, contains $v_{n}$, that is, provided that $d(v_{n}, v) < \varepsilon$. > \ > Equivalently, $(v_{n}) \to v$ iff for every [[neighborhood]] $V$ of $v$ there exists $N \in \mathbb{N}$ s.t. for all $n > N$ we have $x_{n} \in V$. > [!definition] > (see [[converge]]) > [!basicexample] > The sequence $\left( \frac{1}{n} \right)_{n \in \mathbb{N}}$ converges to $0$. This is observed thus: given $\varepsilon>0$, choose $N$ large enough that $\frac{1}{N} < \varepsilon$. Then for any $n < N$ we have $| \frac{1}{n} - 0 |< \frac{1}{N} < \varepsilon$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```