Continuous Functions Preserve Sequence Limits

Last updated Nov 1, 2022

Let $(X, \tau), (Y, \rho)$ be Topological Spaces and suppose $f: X \to Y$ is Continuous Function. Suppose $(x_n) \subset X$. If $x_{n} \to x \in X$ , then $f(x_{n}) \to f(x) \in Y$.

Suppose $V \subset Y$ is Open and $f(x) \in V$. Then $f^{-1}(V)$ is Open in $X$ and $x \in f^{-1}(V)$. Thus, there exists $N \in \mathbb{N}$ so that $\forall n \geq N$ we have that $x_{n} \in f^{-1}(V)$. Thus $\forall n \geq N$ we have $f(x_{n}) \in V$. Since $V$ was arbitrary, $f(x_{n}) \to f(x)$. $\blacksquare$

Follows by noting A Sequence is a Net and A Function is Continuous iff it preserves Net Convergence. $\blacksquare$

• Sequence Convergence