How did Abraham Robinson show that infinitesimals could be used consistently?

In the 1960s, Abraham Robinson demonstrated that infinitesimals could be used consistently by developing nonstandard analysis, a framework that rehabilitated the nonrigorous infinitesimal arguments used in early calculus.12

Robinson’s approach relied on the use of nonstandard models of analysis. His logical argument was based on the principle that if the assumptions regarding the existence of an infinitesimal ($\xi$) were to lead to a contradiction, that contradiction would necessarily be derivable from a finite subset of those assumptions. Because any such finite subset—such as the requirement that $\xi$ be smaller than $1/n$ for a specific integer $n$—is consistent (as it can be satisfied by setting $\xi =1/(n+1)$), the entire set of assumptions must also be consistent.21

By proving that infinitesimals could be introduced as valid, new kinds of numbers, Robinson showed that the intuitive methods of early mathematicians like Gottfried Wilhelm Leibniz were logically sound, often allowing for justifications that were simpler than those provided by standard limit-based calculus.12