# College Math Teaching

## May 24, 2013

### Beware of the limiting process….

Filed under: advanced mathematics, research — collegemathteaching @ 12:49 pm

If you’ve done mathematical research, you are probably aware of the following minefield (especially if you study non-compact spaces):

1. Establish $f_1 = f_2 = ....f_n$ for all $n \in \{1, 2, 3, ...\}$. Note: I am abusing the $=$ sign here; I mean “equivalence class equality”.

2. Then try to conclude that $lim_{n \rightarrow \infty}f_n = f$ only to find out….that the limit fails to exist, though it might exist if you put restrictive conditions on either the $f_n$ or on HOW the equivalence is obtained. 😦

This really shouldn’t surprise me at all; after all one of the things we teach our advanced calculus students is this example:

let $f_0(x) = 1, 0 \le x \le 1$. Index the rational numbers by $q_i$
Let $f_n(x) = 1$ if $x \notin \{q_1, q_2, ... q_n\}$, $f_n(x) = 0$ if $x \in \{q_1, q_2, ...q_n \}$. Then, while it is true that $\int^1_0 f_n(x) dx = 1$ for all $n$, the limit $lim_{n \rightarrow \infty}f_n$ fails to be Riemann integrable (though it is Lebesgue integrable).

How quickly I sometimes forget the basics. 🙂