Re: DSP riddle

Randy Yates wrote:
"Andor" <andor.bariska@xxxxxxxxx> writes:

Another paraphrase is: the set X lies dense in [0,1] if for every real
number x in [0,1] there exists a sequence x_n in X, such that lim_{n ->
infinity} x_n = x. In general, the limit x is not contained in X.

For example, the rational numbers lie dense in the real numbers (every
irrational number can be written as the limit of a sequence of rational
numbers). Or the Weierstrass approximation theorem: the polynomials lie
dense in the continuous functions over the interval [0,1].


Thanks, Andor. I might have to have some more schooling before I can
understand this.

My Calculus teacher liked the "axiomatical" approach to introduce the subject, as opposed to most of his colleagues who preferred the "constructive" one. It seems the problem in the axiomatic approach is axiom #15, the "supreme axiom"... I recall thinking of it over and over :-) hard to see the point.

The problem in the constructive approach is the construction of reals based on rationals. The mathematician who did it was called Cantor ("triangulation of Cantor" is the name of the proof that there are reals which are not rationals) and he became mad after that (Cantor, not my teacher) . My teacher was sad about how much more interesting work Cantor could have done if he had not became obsessed with that particular problem. In the end, the proof grounded in the fact that you can build a sequence of rationals converging to an irrational, this is, it worked because rationals are dense on reals. The supreme theorem said something like any bounded set of reals has supreme... either because it's there on the set (maximum) or because you can build a sequence converging to it. My teacher used to comfort us telling us that Cantor got mad. He was clearly magnifying the problem to motivate us.

The corollary was we shouldn't feel annoyed to get stuck with that stuff. "It took the most brilliant mathematicians to break that wall!!" he'd say :-). A week later I solved an exercise using the supreme axiom and got congratulations from my teacher :-)

-javier
.

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