Sequence of Functions

A sequence of functions is a function whose domain is $\mathbb{N}$ and whose range is a set of functions.

Convergence

Pointwise Convergence

• (d6.1) Given a sequence of functions $(f_n)$, and a function $f$ such that all functions defined on an interval $I$. The following are equivalent:

  • The sequence $(f_n)$ converges pointwise on $I$ to the limit function $f$
  • $\forall x\in I,f(x)=\underset{n\to \infty }{\lim }{f}_n(x)$
  • $f_n\to f$ pointwise on $I$
  • For every $x\in I$, the sequence $(f_n)$ converges at $x$ to $f(x)$
  • $\forall \epsilon >0,\forall x\in I,\exists N\in \mathbb{N}:\forall n\ge N,|f_n(x)-f(x)|<\epsilon$

• If $(f_n)$ converges pointwise to $f$ on $[a,b]$

  • (6.9) term-by-term differentiation - If each $f_n$ is continuously differentiable on $[a,b]$, and if the sequence of derivatives $(f_n^{\prime })$ converges uniformly on $[a,b]$, then $f$ is differentiable on $[a,b]$ and $f^{\prime }(x)=\underset{n\to \infty }{\lim }{f}_n^{\prime }(x)$ for all $x\in [a,b]$.
  • (6.5) Dini’s Theorem - If $f$ is continuous on $[a,b]$, and $(f_n)$ is increasing ($\forall n\in \mathbb{N},\forall x\in [a,b],f_n(x)\le f_{n+1}(x)$), or decreasing, then $(f_n)$ converges uniformly to $f$ on $[a,b]$

Uniform Convergence

• (d6.2, 6.3) Given a sequence of functions $(f_n)$, and a function $f$ such that all functions defined on an interval $I$. The following are equivalent:
  • The sequence $(f_n)$ converges uniformly on $I$ to the uniform limit $f$
  • $f_n\to f$ uniformly on $I$
  • $\forall \epsilon >0,\exists N\in \mathbb{N}:\forall n\ge N,\forall x\in I,|f_n(x)-f(x)|<\epsilon$
  • $f$ is the uniform limit of $(f_n)$ on $I$
  • (6.3) $\underset{n\to \infty }{\lim }\underset{x\in I}{\sup }|f_n(x)-f(x)|=0$
  • (q6.7) There exists a null sequence $(a_n)$ such that for almost all $n$ and for all $x\in I$, we have $|f_n(x)-f(x)|\le a_n$

Cauchy Criterion for UC

• Given a sequence of functions $(f_n)$ such that all functions defined on an interval $I$. The following are equivalent:
  • There exists a function $f$ such that $f_n\to f$ uniformly on $I$
  • ((6.6) Cauchy Criterion for U.C.) $\forall \epsilon >0,\exists N:\forall n>N,\forall p\in \mathbb{N},\forall x\in I,|f_{n+p}(x)-f_n(x)|<\epsilon$

Theorems & Properties

• If $f_n\to f$ uniformly on $I$, then:

  • (q6.5) $f_n\to f$ pointwise on $I$
  • (6.4) If each $f_n$ is continuous on $I$, then $f$ is continuous on $I$.
  • (q6.10) If each $f_n$ is bounded on $I$, then $f$ is bounded on $I$.
  • (q6.11) If $f$ is bounded on $I$, then:
    • There exists a constant $A$ such that for almost all $n$ we have $\underset{I}{\sup }\{\left|f_n(x)\right|\}\le A$ (i.e. from some $n$ onwards, all $f_n$ are bounded by $A$)
    • $\underset{n\to \infty }{\lim }\left(\underset{I}{\sup }\{\left|f_n(x)\right|\}\right)=\underset{I}{\sup }\{\left|f(x)\right|\}$

• If each $f_n$ is integrable on $[a,b]$ and if $(f_n)$ converges uniformly to $f$ on $[a,b]$, then $f$ is integrable on $[a,b]$ and ${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }{\int }_a^b{f}_n(x)dx$.

• If each $f_n$ is differentiable continuously on $[a,b]$ and if the sequence of derivatives $(f_n^{\prime })$ converges uniformly on $I$, and if $(f_n)$ converges at some point $x_0\in I$, then $(f_n)$ converges uniformly on $I$ to a function $f$ that is differentiable on $I$ and $f^{\prime }(x)=\underset{n\to \infty }{\lim }{f}_n^{\prime }(x)$ for all $x\in I$.

• If $f_n\to f$ and $g_n\to g$ uniformly on $I$, then:

  • (q6.9a) $f_n+g_n\to f+g$ uniformly on $I$
  • (q6.9b) if $f$ and $g$ are bounded on $I$, then $f_n\cdot g_n\to f\cdot g$ uniformly on $I$