If I understand correctly all sequences who converge towards a rational limit are self-asymptotic.
Example (dichotomy series):
$\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \frac{31}{32}, \frac{63}{64}...$
$\frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \frac{31}{32}, \frac{63}{64}...$
$\frac{7}{8}, \frac{15}{16}, \frac{31}{32}, \frac{63}{64}...$
The difference between the sequences (which is basically the first one removing terms) is diminishing.
The same property exists for infinite non-periodic decimal series
e.g. $0.10111213141516....$
which can be approximated by the sequence:
$0.1, 0.10, 0.101, 0.1011, 0.10111, 0.101112 ....$
if we remove some terms the new sequence e.g.
$ 0.101112, 0.1011121, 0.10111213, 0.101112131 ....$
will be asymptotic to the above hence self-asymptotic.
What is not clear to me is if the periodic decimal series also holds this self-asymptotic property
E.g.
$0.363636363636....$
It seems to me that it is but I am not sure if we remove terms from the approximating sequence
I.e.
$0.36, 0.363, 0.3636, 0.36363, 0.363636....$
$0.3636, 0.36363, 0.363636....$
in the end they actually reach $0$ and do not converge in the way we define the convergence above.