Topology: An Inquiry-Based Approach

A sequence in a metric space $X$ is a function $f:{\mathbb{Z}}^{+}\to X\text{.}$

A sequence $(a_n)$ in a metric space $(X,d)$ has a limit $L$ in $X$ if, given any $ϵ>0$ there exists a positive integer $N$ such that $d(a_n,L)<ϵ$ whenever $n\ge N\text{.}$

Let $f$ be a function from a metric space $(X,d_X)$ to m metric space $(Y,d_Y)\text{.}$ Then $f$ is continuous at $a\in X$ if and only if $limf(a_n)=f(a)$ for any sequence $(a_n)$ in $X$ that converges to $a\text{.}$