----- > [!proposition] Proposition. ([[continuity preserves convergence]]) > Suppose that $(a_{n})_{n \in \mathbb{N}}$ is a [[converge|convergent]] [[sequence]] in a [[topological space]] $X$ and $f:X \to Y$ is a [[continuous]] function. Then $\big( f(a_{n}) \big)_{n \in \mathbb{N}}$ is a [[converge|convergent]] [[sequence]] in $Y$. > \ > In particular, if $a \in X$ is a point to which $(a_{n})_{n \in \mathbb{N}}$ converges, then $f(a)$ is a point to which $\big( f( a_{n}) \big)_{n \in \mathbb{N}}$ converges. > [!proof]- Proof. ([[continuity preserves convergence]]) > ~ > Let $a \in X$ be a point to which $(a_{n})_{n \in \mathbb{N}}$ [[converge|converges]]. We will show that $(f(a_{n}))_{n \in \mathbb{N}}$ [[converge|converges]] to $f(a)$. > Let $V$ be an open [[neighborhood]] containing $f(a)$. Since $U:=f^{-1}(V)$ is an open [[neighborhood]] of $a$, there exists $N \in \mathbb{N}$ s.t. for all $n>N$ we have $a_{n} \in U$. It follows that for all $n > N$ we have $f(a_{n}) \in f(U)=V$. This is the result. ----- #### ---- #### 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 > ```