Sequences of Functions Review

We will now review some of the recent material regarding sequences of functions. Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of functions with common domain $X$.

• On the Sequences of Functions we defined a Sequence of Functions denoted $(f_n(x))_{n=1}^{\infty}$ where $f_n(x)$ is a function for each $n \in \mathbb{N}$. We looked at a couple examples of sequences of functions such as the sequence $(f_n(x))_{n=1}^{\infty} = (nx)_{n=1}^{\infty} = (x, 2x, 3x, ...)$ of diagonal lines with increasing slope.

• On the Pointwise Convergence of Sequences of Functions we said that $(f_n(x))_{n=1}^{\infty}$ is Pointwise Convergent to the function $f(x)$ denoted $\displaystyle{\lim_{n \to \infty} f_n(x) = f(x) \: \mathit{pointwise}}$ if for all $x \in X$ and for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then:

(1)

\begin{align} \quad \mid f_n(x) - f(x) \mid < \epsilon \end{align}

• We looked at an example of determining pointwise convergence of a sequence of functions on the Determining Pointwise Convergence of Sequences of Functions page.

• Similarly on the Uniform Convergence of Sequences of Functions page we said that $(f_n(x))_{n=1}^{\infty}$ is Uniformly Convergent to the function $f(x)$ denoted $\displaystyle{\lim_{n \to \infty} f_n(x) = f(x) \: \mathit{uniformly}}$ if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then for all $x \in X$ we have that:

(2)

\begin{align} \quad \mid f_n(x) - f(x) \mid < \epsilon \end{align}

• On the Pointwise Cauchy Sequences of Functions page we said that $(f_n(x))_{n=1}^{\infty}$ is Pointwise Cauchy if for all $x \in X$ and for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ we have that:

(3)

\begin{align} \quad \mid f_m(x) - f_n(x) \mid < \epsilon \end{align}

• We then proved that a sequence of functions $(f_n(x))_{n=1}^{\infty}$ is pointwise Cauchy if and only if it is pointwise convergent.

• Similarly on the Uniformly Cauchy Sequences of Functions page we said that $(f_n(x))_{n=1}^{\infty}$ if Uniformly Cauchy if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then for all $x \in X$ we have that:

(4)

\begin{align} \quad \mid f_m(x) - f_n(x) \mid < \epsilon \end{align}

• We then proved that a sequence of functions $(f_n(x))_{n=1}^{\infty}$ is uniformly Cauchy if and only if it is uniformly convergent.

• On the Continuity of a Limit Function of a Uniformly Convergent Sequence of Functions page we saw that if each $f_n$ is continuous at a point $c \in X$ and if $(f_n(x))_{n=1}^{\infty}$ uniformly converges to $f(x)$ then we must have that $f$ is continuous at $c$.

• On the Differentiation and Uniformly Convergent Sequences of Functions page we saw that if $(f_n(x))_{n=1}^{\infty}$ is a sequence of differentiable functions on $(a, b)$ where $(f_n'(x))_{n=1}^{\infty}$ uniformly converges to some function $g$ on $(a, b)$ and if there exists an $x_0 \in (a, b)$ such that the numerical sequence $(f_n(x_0))_{n=1}^{\infty}$ converges then $(f_n(x))_{n=1}^{\infty}$ uniformly converges on $(a, b)$ to some function $f(x)$ and moreover, $f$ is differentiable with $f'(x) = g(x)$ for all $x \in (a, b)$.

• On the Riemann Integrability of the Limit Function of a Uniformly Convergent Sequence of Functions page we saw that if each $f_n$ is Riemann integrable on a closed and bounded interval $[a, b]$ and if $(f_n(x))_{n=1}^{\infty}$ uniformly converges to $f(x)$ then we must have that $f$ is Riemann integrable on $[a, b]$ and moreover:

(5)

\begin{align} \quad \int_a^b f(x) \: dx = \int_a^b \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_a^b f_n(x) \: dx \end{align}

• We then extended this notion on the Uniform Convergence of Sequences of Functions and Riemann-Stieltjes Integration Part 1 and Uniform Convergence of Sequences of Functions and Riemann-Stieltjes Integration Part 2 page. We saw that if $\alpha$ is a function of bounded variation on $[a, b]$, each $f_n$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$, and if $(f_n(x))_{N=1}^{\infty}$ uniformly converges to $f(x)$ then we must have that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and moreover:

(6)

\begin{align} \quad \int_a^b f(x) \: d \alpha(x) = \int_a^b \lim_{n \to \infty} f_n(x) \: d \alpha (x) = \lim_{n \to \infty} \int_a^b f_n(x) \: d \alpha (x \end{align}