Content deleted Content added
Line 216:
I agree that this can be a subtle point. The important point is to distinguish what Cantor's theorem does in general from what it does in a particular application, such as being applied to the sequence of all algebraic numbers in an interval. In general, Cantor's theorem only guarantees that the real number obtained from his construction is not in the given sequence. In many applications, such as applying it to the sequence of rationals in (0, 1) as in the section "Example of Cantor's construction", you are not guaranteed to obtain a transcendental number but in this case you are guaranteed to obtain an number that is not rational. Applying the theorem to this case, you obtain <math>\sqrt{2}-1</math>, which is a non-rational algebraic number.
However, if you apply Cantor's theorem to the sequence of all algebraic numbers in an interval (''a'', ''b''), you are guaranteed to obtain a non-algebraic number in the interval.
Since you brought up that you can hit an algebraic number in Case 1 or Case 3, I wish to point out that the section "Dense sequences" proves that a dense sequence—such as, the sequence of algebraic numbers in an interval—never ends up in Case 1 or Case 3, so your argument fails here. To obtain a transcendental number, Cantor is using such a dense sequence.
| |||