In the book on Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, p.4) there is such term as "standard infinitesimals". This infinitesimals can be represented as Cauchy sequence following way:
$x(t)=y(t)+o(t)$ for $t\to0$
Let $x(t)=t$ and $y(t)=\sin(t)$.
Cauchy sequences: $(0.1, 0.01, 0.001, ...) = (0.0998..., 0.0099..., 0.0009..., ...) + o(t)$
Thus $(0.1, 0.01, 0.001, ...) = (0.0998..., 0.0099..., 0.0009..., ...)$ and this equivalent sequences represent infinitesimal number $h=[t]=[\sin(t)]$.
But $[t]=(0.1, 0.01, 0.001, 0.0001, ... )$ represents $0$.
How is it possible that Cauchy sequence represents two different numbers?
Thanks.