How did Karl Weierstrass give a rigorous definition of a limit?

Before the 19th century, mathematicians relied on intuitive, often vague concepts of limits involving "infinitesimals" or "fluxions," which faced significant criticism for lacking logical rigor. Karl Weierstrass resolved these issues by providing a formal, precise definition that eliminated reliance on mystical notions of infinity.123

Weierstrass defined the limit of a sequence $(a_n)$ as follows: the sequence converges to a limit $a$ if, for every desired degree of closeness $ϵ>0$, there exists a whole number $N$ such that the difference between the sequence terms and the limit is less than $ϵ$ for all $n>N$. Mathematically, this is expressed as:

$$|a_n-a|<ϵ\text{ for all }n>N$$

This definition ensures that for any chosen error margin $ϵ$, one can find a point in the sequence beyond which all subsequent terms remain within that distance of the limit. By grounding the concept entirely in real numbers rather than infinitesimals, Weierstrass established a rigorous foundation for analysis.12