Pointwise Convergence

Definition: #

  Let $\bf \{ f_{n}(x)\}$ be a sequence of real velued functions, defined on $\bf E\subseteq \real$. The sequence $\bf \{ f_{n}(x)\}$ is said to be $\fcolorbox{red}{white}{\bf pointwise convergent}$to a real valued function $\bf f(x)$, defined on $\bf E\subseteq \real$, if for any positive number $\bf \epsilon~,~\exists$ a positive integer $\bf k$ such that $$\begin{array}{rl}{\displaystyle }& {\displaystyle \mathbf{\mid }{\mathbf{f}}_{\mathbf{n}}(\mathbf{x})-\mathbf{f}(\mathbf{x})\mathbf{\mid }<\mathbf{ϵ}\mathbf{\forall }\mathbf{n}\ge \mathbf{k}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{x}\in \mathbf{E}}\\ & {\displaystyle }& {\displaystyle }& & \end{array}$$$$\begin{array}{rl}{\displaystyle }& {\displaystyle \mathbf{\mid }{\mathbf{f}}_{\mathbf{n}}(\mathbf{x})-\mathbf{f}(\mathbf{x})\mathbf{\mid }<\mathbf{ϵ}\mathbf{\forall }\mathbf{n}\ge \mathbf{k}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{x}\in \mathbf{E}}\\ & {\displaystyle }& {\displaystyle }& & \end{array}$$$$\begin{array}{rl}{\displaystyle }& {\displaystyle \mathbf{\mid }{\mathbf{f}}_{\mathbf{n}}(\mathbf{x})-\mathbf{f}(\mathbf{x})\mathbf{\mid }<\mathbf{ϵ}\mathbf{\forall }\mathbf{n}\ge \mathbf{k}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{x}\in \mathbf{E}}\\ & {\displaystyle }& {\displaystyle }& & \end{array}$$$$\begin{align} & \bf|f_{n}(x)-f(x)|<\epsilon~\forall~n\ge k~and~x\in E \nonumber \\ & \end{align}$$   And is denoted by $$\bf \lim\limits_{n\to \infty}f_{n}(x)=f(x)$$

 Example-1: $\bf f_{n}(x)=x^{n}$ on $\bf x\in [0,1]$ #

 Solution:

  We know that the sequence of real number $\bf \{y_{n} \}$, where $\bf y_{n}=x^{n}~\forall~|x|\text{\textless} 1$, converges to $\bf 0$. Then $$\bf \lim\limits_{n\to \infty}y_{n}=0$$ Thus we have $$\bf \lim\limits_{n\to \infty}f_{n}(x)=0~when~0\le x \text{\textless} 1$$  Also, for $\bf x=1$, we have the sequence $\bf \{ f_{n}(1) \}=\{1,1,….\}$. Therefore we have $$\bf \lim\limits_{n\to \infty}f_{n}(x)=1~when~x=1$$  Hence $$\bf \lim\limits_{n\to \infty}f_{n}(x)=f(x)~when~x\in [0,1]$$ where $$\bf f(x)=\begin{cases} \bf 0 &\text{\bf if } \bf x\in [0,1) \\ \bf 1 &\text{\bf if } \bf x=1 \end{cases}$$