Pointwise and uniform convergenceThe aim here is to show some videos illustrating the difference between [b]pointwise[/b] and [b]uniformly[/b] convergent sequences of functions.[b]Example 0:[/b] [img]https://s0.wp.com/latex.php?latex=f_n%28x%29+%3D+x%5En&bg=f9f9f9&fg=555555&s=0&c=20201002[/img] at the point just gives the sequence [img]https://s0.wp.com/latex.php?latex=a_n+%3D+2%5En&bg=f9f9f9&fg=555555&s=0&c=20201002[/img]. This viewpoint (ie, sequences arising from fixed x values) is shown in the left picture below. However we normally consider the sequence of functions as [img]https://s0.wp.com/latex.php?latex=n&bg=f9f9f9&fg=555555&s=0&c=20201002[/img] increases, and this is [br]shown in the figure below right.[br][br][br][url=https://zoewyatt.wordpress.com/point-unif-convergence/f4/][img]https://zoewyatt.files.wordpress.com/2019/02/f4.png?w=1024&h=584[/img][br][br][br][br][/url][url=https://zoewyatt.wordpress.com/point-unif-convergence/f3/][img]https://zoewyatt.files.wordpress.com/2019/02/f3-1.png?w=1024&h=590[/img][br][br][br][br][/url][b]Example 1:[/b] [img]https://s0.wp.com/latex.php?latex=f_n%28x%29+%3D+%5Cfrac%7B%5Csin%28nx%2B3%29%7D%7B%5Csqrt%7Bn%2B1%7D%7D&bg=f9f9f9&fg=555555&s=0&c=20201002[/img] in blue converges pointwise (and in fact uniformly) to [img]https://s0.wp.com/latex.php?latex=f%28x%29+%3D+0&bg=f9f9f9&fg=555555&s=0&c=20201002[/img] in orange [br][br]below.[img]https://zoewyatt.files.wordpress.com/2019/02/f2-disc-2.gif?w=1108[/img]

[size=150]Uniform and pointwise convergence are two different ways to describe the behavior of a sequence of functions converging to a limiting function. Let's explain both concepts using an example.[br][/size][br]E[size=150]xample: Consider the sequence of functions[/size] [math]fn\left(x\right)=\frac{1}{n}\left(x\right)[/math] [size=150]defined on the interval [/size][0, 1].[br][br]1.[size=150] Pointwise Convergence:[br]Pointwise convergence refers to the convergence of the individual function values at each point in the domain as (n) approaches infinity.[br][br]For this example, pointwise convergence means that for each (x) in the interval [0, 1], we examine the behavior of f[sub]n[/sub](x) as (n) becomes very large.[/size][br][br]f[sub]n[/sub](x) = [math]\frac{1}{n}\left(x\right)[/math][br][br][size=150]As (n) increases, the value of f[sub]n[/sub](x) approaches 0 for any fixed (x) in the interval [0, 1]. It means that as (n) becomes large, each function f[sub]n[/sub](x) approaches the zero function.[br][br]2. Uniform Convergence:[br]Uniform convergence refers to the convergence of the entire sequence of functions as a whole, rather than just the individual function values. It means that the difference between f[sub]n[/sub](x) each and the limiting function (f(x)) becomes arbitrarily small for all points in the domain as (n) approaches infinity.[br][br]For this example, the limiting function is (f(x) = 0) (the zero function).[br][br]We say that the sequence of functions f[sub]n[/sub](x) = [math]\frac{1}{n}\left(x\right)[/math] converges uniformly to the zero function (f(x) = 0) on the interval [0, 1] if, for any [math]\epsilon[/math]>0, there exists an [i]N[/i] such that for all (n > N) and for all (x) in [0, 1], the inequality [math]\left|fn\left(x\right)-f\left(x\right)\right|[/math]<[math]\epsilon[/math] holds.[br][math][/math][br]In this example, the sequence of functions f[sub]n[/sub](x) = [math]\frac{1}{n}\left(x\right)[/math] converges uniformly to the zero function on [0, 1] because for any [math]\epsilon[/math] > 0, we can always choose [math]\mathbb{N}=\frac{1}{\epsilon}[/math]. Then, for all (n > N) and for all (x) in [0, 1], we have:[br][br][math]\left|fn\left(x\right)-f\left(x\right)\right|=\left|\frac{1}{n}\left(x\right)-0\right|=\frac{1}{n}\left(x\right)<\frac{1}{N}\left(x\right)<\frac{1}{N}\le\in[/math][br][br]So, the difference between f[sub]n[/sub](x) and (f(x)) becomes arbitrarily small for all (x) in the interval [0, 1] as (n) becomes large.[br][br]In summary, pointwise convergence considers the behavior of individual function values at each point, while uniform convergence ensures that the entire sequence of functions approaches the limiting function in a uniform manner across the entire domain. In this example, the sequence of functions f[sub]n[/sub](x) = [math]\frac{1}{n}\left(x\right)[/math] converges pointwise to the zero function (f(x) = 0), but it converges uniformly to the same zero function on the interval [0, 1].[/size]