Uniform Convergence of Sequences of Functions

Sequences of Functions

Definition: An Infinite Sequence of Functions $(f_{n}(x))_{n=1}^{\infty }=(f_{1}(x),f_{2}(x),...,f_{n}(x),...)$ is a sequence of functions with a common domain. The $n^{\mathrm {th} }$ Term of the sequence is the function $f_{n}(x)$.

We can define a finite sequence of functions analogously. A finite sequence of functions is denoted $(f_{n})_{n=1}^{\infty }$.

We can also denote an infinite sequence of functions as simply $(f_{n}(x))$. We can also use curly brackets to denote a sequence of functions such as $\{f_{n}(x)\}_{n=1}^{\infty }$ or simply $\{f_{n}(x)\}$.

For example, consider the following sequence of functions:

${\begin{aligned}\quad (f_{n}(x))_{n=1}^{\infty }=\left(nx\right)_{n=1}^{\infty }=(x,2x,...,nx,...)\end{aligned}}$

This is a sequence of diagonal straight lines that pass through the origin and whose slope is increasing. The following illustrates a few of the functions in this sequence:

Graph of a Sequence of Linear Functions

For another example, consider the following sequence of functions:

${\begin{aligned}\quad (f_{n}(x))_{n=1}^{\infty }=(x^{n})_{n=1}^{\infty }=(x,x^{2},x^{3},...,x^{n},...)\end{aligned}}$

This is a sequence of the simplest $n^{\mathrm {th} }$ degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:

Graph of a Sequence of Polynomials

Pointwise Convergence of Functions

Definition: Let $(f_{n}(x))_{n=1}^{\infty }$ be a sequence of functions with common domain $X$. Then $(f_{n})_{n=1}^{\infty }$ is said to be Pointwise Convergent to the the function $f$ written $\lim _{n\to \infty }f_{n}(x)=f(x)$ if for all $x\in X$ and for all $\varepsilon >0$ there exists a $N\in \mathbb {N}$ such that if $n\geq N$ then $\mid f_{n}(x)-f(x)\mid <\varepsilon$.

For example, consider the following sequence of functions defined on $[0,1]$:

${\begin{aligned}\quad (f_{n}(x))_{n=1}^{\infty }=\left({\frac {1}{n}}x\right)_{n=1}^{\infty }=\left(x,{\frac {1}{2}}x,{\frac {1}{3}}x,...,{\frac {1}{n}}x,...\right)\end{aligned}}$

We claim that $(f_{n}(x))_{n=1}^{\infty }$ is pointwise convergent to $f(x)=0$. The following image shows the first six functions in the sequence given above. It should be intuitively clear that the sequence $(f_{n}(x))_{n=1}^{\infty }$ converges to the limit function $f(x)=0$.

To show this, fix $x\in [0,1]$ and assume that $x\neq 0$ and let $\varepsilon >0$ be given. Then since $n,x>0$ we have that:

${\begin{aligned}\quad \left|f_{n}(x)-f(x)\right|=\left|{\frac {1}{n}}x-0\right|=\left|{\frac {x}{n}}\right|={\frac {x}{n}}\end{aligned}}$

Choose $N\in \mathbb {N}$ such that $N>{\frac {x}{\varepsilon }}$ which can be done by the Archimedean property. Then ${\frac {1}{N}}<{\frac {\varepsilon }{x}}$ and so for $n\geq N$ we have that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \frac{x}{n} \leq \frac{x}{N} < \frac{\varepsilon x}{x} = \varepsilon \end{align}}

Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} f_n(x) = f(x)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in (0, 1]} . Now, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0} , notice that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (f_n(0))_{n=1}^{\infty} = (0)_{n=1}^{\infty} = (0, 0, ...) \end{align}}

This sequence clearly converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0) = 0} . So, we conclude that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\lim_{n \to \infty} f_n(x) = f(x)}} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in [0, 1]} . Hence the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} is pointwise convergent on all of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1]} .

Uniform Convergence of Sequences of Functions

The sequence of functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} with common domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is convergent to the limit function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} if for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} and for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \in \mathbb{N}} such that if $n\geq N$ then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid f_n(x) - f(x) \mid < \varepsilon} .

Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!

Definition: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} be a sequence of functions with common domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} is said to be Uniformly Convergent to the the limit function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} (written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} f_n(x) = f(x)} uniformly on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} or as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n \to f } uniformly on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ) if for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \in \mathbb{N}} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq N} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid f_n(x) - f(x) \mid < \varepsilon} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} .

Graphically, if the sequence of functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} are all real-valued and uniformly converge to the limit function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , then from the definition above, we see that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \in \mathbb{N}} such that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq N} we have that the following inequality holds for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad f_n(x) - \varepsilon < f(x) < f_n(x) + \varepsilon \end{align}}

The following graphic illustrates the concept of uniform convergence of a sequence of functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} :

Uniform Convergence of a Sequence of Functions

Licensing

Content obtained and/or adapted from:

• Sequences of Functions, mathonline.wikidot.com under a CC BY-SA license
• Pointwise Convergence of Sequences of Functions under a CC BY-SA license
• Uniform Convergence of Sequences of Functions under a CC BY-SA license