# College Math Teaching

## April 17, 2012

### Pointwise versus Uniform convergence of sequences of continuous functions: Part II

Filed under: analysis, calculus, complex variables, uniform convergence — collegemathteaching @ 12:48 am

In my complex analysis class I was grading a problem of the following type:
given $K \subset C$ where $K$ is compact and given a sequence of continuous functions $f_n$ which converges to 0 pointwise on $K$ and if $|f_1(z)| > |f_2(z)|>...|f_k(z)|...$ for all $z \in K$ show that the convergence is uniform.

Now what about the $|f_1(z)| > |f_2(z)|>...|f_k(z)|...$ hypothesis? Can it be dispensed with?

Let’s look at an example in real variables:

Let $f_n(x) = sin(\frac{e \pi}{2}e^{-nx})sin(\frac{n \pi}{2} x)$ with $x \in [0,1]$. $f_n(0) = 0$ for all $n$. To see that $f_n$ converges to zero pointwise, note that $lim_{n \rightarrow \infty}e^{-nx} = 0$ for all $x > 0$, hence $lim_{n \rightarrow \infty}sin(\frac{e \pi}{2}e^{-nx}) = 0$ which implies that $f_n \rightarrow 0$ by the squeeze theorem. But $f_n$ does not converge to 0 uniformly as for $t = \frac{1}{n}$ we have $f_n(t) = 1$

Here is a graph of the functions for $n = 5, 10, 20, 40$