00:00 Intro

00:33 Pointwise convergence

02:03 1st Example

03:20 2nd Example

06:40 3rd Example

07:43 Credits

Q1: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: I \rightarrow \mathbb{R}$. What is the correct definition for the pointwise convergence to a function $f: I \rightarrow \mathbb{R}$?

A1: $ \displaystyle \forall x \in I ~~ \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $

A2: $ \displaystyle \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall x \in I ~~ \forall n \geq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $

A3: $ \displaystyle \forall x \in I ~~ \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \geq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $

A4: $ \displaystyle \forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall x \in I ~~ \forall n \leq N ~ : ~ |f_n(x)-f(x)|<\varepsilon $

Q2: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: [0,1] \rightarrow \mathbb{R}$ given by $f_n(x) = \frac{x}{n^2} - 2 - \frac{1}{n} $. What is the pointwise limit $f$?

A1: $ \displaystyle f(x) = \frac{x}{n^2} - 2 - \frac{1}{n} $

A2: $ \displaystyle f(x) = 0 $

A3: $ \displaystyle f(x) = x - 2 $

A4: $ \displaystyle f(x) = \frac{x}{2} - 2 - \frac{1}{2} $

A5: $ \displaystyle f(x) = -2 $

Q3: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: [0,1] \rightarrow \mathbb{R}$. Which example does not have a pointwise limit?

A1: $ \displaystyle f_n(x) = \frac{x}{n^2} - 2 - \frac{1}{n} $

A2: $ \displaystyle f_n(x) = x - 2$

A3: $ \displaystyle f_n(x) = \begin{cases} n^2 x - 2 &, ~~ x = 0\ x - \frac{1}{n} &, ~~x > 0\end{cases} $

A4: $ \displaystyle f_n(x) = \frac{n^2 x}{2} - 2 $