Uniform Convergence of Sequences of Functions

# Uniform Convergence of Sequences of Functions

Recall from the Pointwise Convergence of Sequences of Functions page that we say the sequence of functions $(f_n(x))_{n=1}^{\infty}$ with common domain $X$ is convergent to the limit function $f(x)$ 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 $\mid f_n(x) - f(x) \mid < \epsilon$.

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 $(f_n(x))_{n=1}^{\infty}$ be a sequence of functions with common domain $X$. Then $(f_n(x))_{n=1}^{\infty}$ is said to be Uniformly Convergent to the the limit function $f$ written $\lim_{n \to \infty} f_n(x) = f(x) \: \mathit{uniformly \: on} \: X$ or $f_n \to f \: \mathit{uniformly \: on} \: X$ if for all $\epsilon > 0$ there exists a $N \in \mathbb{N}$ such that if $n \geq N$ then $\mid f_n(x) - f(x) \mid < \epsilon$ for all $x \in X$.

Graphically, if the sequence of functions $(f_n(x))_{n=1}^{\infty}$ are all real-valued and uniformly converge to the limit function $f$, then from the definition above, we see that for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that for all $n \geq N$ we have that the following inequality holds for all $x \in X$:

(1)
\begin{align} \quad f_n(x) - \epsilon < f(x) < f_n(x) + \epsilon \end{align}

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