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 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^{\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:

<img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.44.29%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.44.29%20PM.png" class="image" />

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

<img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" class="image" />

Pointwise Convergence of Functions

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 $X$. 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)_{n=1}^{\infty}} is said to be Pointwise Convergent to the the 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 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)} if for all $x\in X$ 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 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 example, consider the following sequence of functions defined on $[0,1]$:

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))_{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{align}}

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

To show this, fix 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]} and assume 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 x \neq 0} and 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 \varepsilon > 0} be given. Then since $n,x>0$ 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 \left|f_n(x) - f(x)\right| = \left|\frac{1}{n} x - 0 \right| = \left|\frac{x}{n} \right| = \frac{x}{n} \end{align}}

Choose 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 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 > \frac{x}{\varepsilon}} which can be done by the Archimedean property. 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 \frac{1}{N} < \frac{\varepsilon}{x}} and so 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 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 $\lim _{n\to \infty }f_{n}(x)=f(x)$ 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 $\displaystyle {\lim _{n\to \infty }f_{n}(x)=f(x)}$ 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

Recall from the <a href="/pointwise-convergence-of-sequences-of-functions">Pointwise Convergence of Sequences of Functions</a> page that we say 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 $x\in X$ 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 $\mid f_{n}(x)-f(x)\mid <\varepsilon$.

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 $(f_{n}(x))_{n=1}^{\infty }$ 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 "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)/math> uniformly on <math>X} " or "$f_{n}\to f{\text{ uniformly on }}X$") 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 $n\geq N$ 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 $(f_{n}(x))_{n=1}^{\infty }$ 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 $N\in \mathbb {N}$ 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} :

${\begin{aligned}\quad f_{n}(x)-\varepsilon <f(x)<f_{n}(x)+\varepsilon \end{aligned}}$

The following graphic illustrates the concept of uniform convergence of a sequence of functions <math>(f_n(x))_{n=1}^{\infty}<math>:

<img src="http://mathonline.wdfiles.com/local--files/uniform-convergence-of-sequences-of-functions/Screen%20Shot%202015-10-19%20at%209.25.38%20PM.png" alt="Screen%20Shot%202015-10-19%20at%209.25.38%20PM.png" class="image" />

Licensing

Content obtained and/or adapted from:

• Sequences of Functions, mathonline.wikidot.com under a CC BY-SA license
• [1] under a CC BY-SA license
• [2] under a CC BY-SA license