# Proof of theorem about continuity

$\textbf{4.2}\,\,$ Theorem $\,\,$ Let $X,Y,E,f$, and $p$ be as in Definition $4.1$. Then $$\lim_{x\to p}f(x)=q\tag{4}$$ if and only if $$\lim_{n\to\infty}f(p_n)=q\tag{5}$$ for every sequence $\{p_n\}$ in $E$ such that $$p_n\ne p,\quad\lim_{n\to\infty}p_n=p.\tag{6}$$ *Proof*$\quad$ Suppose $\text{(4)}$ holds. Choose $\{p_n\}$ in $E$ is satisfying $\text{(6)}$. Let $\varepsilon>0$ be given. Then there exists $\delta>0$ such that $d_Y(f(x),q)<\varepsilon$ if $x\in E$ and $0<d_X(x,p)<\delta$. Also, there exists $N$ such that $n>N$ implies $0<d_X(p_n,p)<\delta$. Thus, for $n>N$, we have $d_Y(f(p_n),q)<\varepsilon$, which shows that $\text{(5)}$ holds.