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. πŸ™‚

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