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How did Karl Weierstrass help establish rigorous foundations for analysis?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

Karl Weierstrass, often called the father of modern analysis, was instrumental in establishing rigorous foundations for the field by replacing intuitive concepts with precise, formal definitions.1

His primary contribution was the program known as the arithmetization of analysis, which sought to base the entire discipline on a rigorous development of the real number system. Central to this effort was his formalization of the limit of a sequence. Weierstrass defined the convergence of a sequence $(a_{n})$ to a limit $a$ as follows: for every $ϵ > 0$ , there exists a whole number $N$ such that $[1] | a_{n} - a | < ϵ$ for all $n > N$ .2

This definition was revolutionary because it:
* Eliminated reliance on vague or "mystical" notions of infinitesimals.2
* Relied exclusively on ordinary real numbers.2
* Provided a precise, unambiguous standard for mathematical convergence.2

Weierstrass further demonstrated his commitment to rigor by developing a function that was continuous everywhere but differentiable nowhere, challenging the reliance on geometric intuition that had previously dominated the field.1