Continuity via sequences

Continuity via sequences: Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $f:X\to Y$. Then $f$ is continuous at $x\in X$ if and only if for every sequence $(x_n)$ in $X$, $x_n\to x \quad\Longrightarrow\quad f(x_n)\to f(x).$

This is one of the most-used characterizations of continuity in analysis: it converts an $\varepsilon$–$\delta$ condition into a limit-preservation property.

Proof sketch (optional): If $f$ is continuous, apply the $\varepsilon$–$\delta$ definition to the tail of the sequence. Conversely, if $f$ is not continuous at $x$, build a sequence $x_n\to x$ with $f(x_n)$ staying a fixed distance away from $f(x)$ by choosing $x_n$ in shrinking neighborhoods where the $\varepsilon$–$\delta$ condition fails.