How to correctly choose number for Cauchy sequence for infinitesimal functions? Will it be $x=y=[sin(t)]$ or $x=y=[t]$?

Question

I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions:

If $x(t)=y(t)+o(t)$ for $t$->$0$ then $x=y$ ($x$ and $y$ are numbers which corresponds for Cauchy sequence for functions $x(t)$ and $y(t)$)

For example: if $sin(t)=t+o(t)$ $t$->$0$ then $x=[sin(t)]$ and $y=[t]$ are equal numbers.

But I am confused when I choose number for Cauchy sequence.

My question is: How to correctly choose number for Cauchy sequence for infinitesimal functions? Will it be $x=y=[sin(t)]$ or $x=y=[t]$?

Thanks!