Definition
1
[2DT]Consider a set \(A\), a function \(f:A\to {\mathbb {R}}\) and a sequence of functions \(f_ n:A\to {\mathbb {R}}\). We will say that \(f_ n\) converges to \(f\) pointwise if
\[ \forall x\in A~ ,~ \lim _{n\to \infty }f_ n(x) = f(x)\quad . \]
We will say that \(f_ n\) converges to \(f\) uniformly if
\[ \forall \varepsilon {\gt}0 \exists N\in {\mathbb {N}}, \forall n \ge N,\forall x\in A~ ,~ |f_ n(x) - f(x)|{\lt}\varepsilon \quad . \]