Pointwise convergence

Definition

Let $S$ be a set, and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$ and $f \colon S \to \mathbb{R}$ be functions. The sequence $(f_n)$ converges pointwise to $f$ on $S$ if $\lim_{n \to \infty} f_n(x) = f(x)$ for every $x \in S$. That is, for every $x \in S$ and every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$.