Prove that $D$ is an ideal of $^\bullet \mathbb R$ using limsup

Question

I read about model construction in synthetic differential geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf)

I try to understand Theorem 1.1 and its proof, but I can't.

Can anyone explain why do we use limsup to prove that $D$ is an ideal and every element of $D$ has square equal to zero?

Thanks.