Date: Nov 25, 2012 4:37 PM
Subject: Matheology § 162
Matheology § 162
About limits of real sequences.
The limit of an infinite sequence (a_k) of real numbers a_k is
determined solely by the finite terms of the sequence. Otherwise, the
limit would not have to be *computed* but would have to be *created*.
Analysis is concerned with analyzing, i.e., with finding.
To give an example, we can state with absolute certainty that in the
real numbers the sequence
0.1, 0.11, 0.111, ...
has the limit 0.111... = 1/9.
That is independent of the method which is used to analyze the
sequence. But there are different aspects of the limit, namely the
numerical value of the limit, the set of coefficients of the power
series, its cardinal number, the set of indexes which belong to a
digit 1, its cardinal number, the set of indexes which belong to a
digit 2, its cardinal number, the set of different digits appearing in
the limit, and many further aspects.
If any of these aspects is computed by another than the analytical
method and turns out as deviating from the analytical result, then the
other method is not suitable for analytical purposes.