Dedekind defining real numbers as equivalence classes of Cauchy sequences of rational numbers. $x=y$ means $x-y=0$ ie $x_n - y_n \to 0$. addition and multiplication are defined for each coordinate.
motivated by continued fractions. what if we rescale so that $x=y$ means $n^2(x_n - y_n) \to 0$ does the resulting number system still lead to the real numbers $\mathbb{R}$ or is there extra "fuzz" around 0?
this is motivated by the result that $x$ is irrational iff there are infinitely many fractions such that
$$ \left| x - \frac{p_n}{q_n} \right| < \frac{1}{q_n^2}$$
The standard construction of the real numbers via Cauchy sequences can be understood algebraically as follows: Let $R$ be the ring of Cauchy sequences of rational numbers under the operations of component-wise addition and multiplication. Then $\mathfrak m = \{(x_n)_{n \in \mathbb N} \in R \mid x_n \to 0\}$ is a maximal ideal of $R$, and we define $\mathbb R = R/\mathfrak m$.
Your proposed number system uses instead the ideal $I = \{(x_n)_{n \in \mathbb N} \in R \mid n^2 x_n \to 0\}$; it is easy to verify that $I$ is indeed an ideal in $R$, but $I \subsetneq \mathfrak m$, so $R/I$ is not a field. Moreover, $r = (1/n^2)_{n \in \mathbb N} \notin I$, but $r^2 \in I$, so $R/I$ isn't even an integral domain.
So in answer to your question, yes, there is "fuzz" around $0$. I don't know whether or not $\mathfrak m/I$ could be seen as an "interesting" system of infinitesimals, but it definitely behaves differently from, say, the system of infinitesimals in the hyperreals, since you have nilpotents.