## Monday, April 18, 2011

### in which annoyance leads to clarification ..

argh! most of my students think that every bounded sequence of real numbers converges!?! .. is this some sort of temporary insanity, caused by quizzes?

[during lecture, last week friday]:

"before we go into the bοlzano-weιerstrass theorem, there are a few things i should clear up. first of all, what's wrong with the following proof?
"since the equation
$$1 = (-1)^{2n} = (-1)^n(-1)^n$$
"holds true for all $n \in \mathbb{N}$, it follows that
$$\lim_{n \to \infty} 1 \;=\; \left( \lim_{n \to \infty} (-1)^n \right) \left( \lim_{n \to \infty} (-1)^n \right).$$
"because the right hand side limits don't exist, it follows that constants sequences such as $x_n = 1$ are actually divergent, not convergent."
hopefully they got the point. \-:

on an unrelated note,
1. it was another busy weekend of technical details .. and other things;
2. these days I use the phrase "the following" a lot. i blame my writing habits.