Definition (Pointwise convergence)

Let $I\subset \mathbb{R}$ be an any interval and let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of real-valued functions on $I$:

1. The sequence converges pointwise on $I$ if the limit $$\lim_{ n \to \infty } f_{n}(x)$$ exists for each point $x\in I$
2. The series $$\sum_{n=1}^{\infty}f_{n}(x)$$converges pointwise on $I$ is the series Convergence for each point $x\in I$
3. $(f_{n})_{n\in\mathbb{N}}$ converges pointwise on $I$ iff $$\forall\epsilon>0,\ \forall x\in I,\ \exists N>0 :|f_{n}(x)-f(x)|<\epsilon\ \forall n\ge N$$