# Real life applications of Cauchy Sequences?

What are some examples of real life applications of Cauchy sequences?(examples that could be modeled.) I read somewhere that the long term effects of weather on a construction could be modeled by a Cauchy sequence. Is it true? How is there convergence? Or could you give any other examples?

• The importance of Cauchy sequences: in a complete space Cauchy $\implies$ convergent. And in the Cauchy condition the limit is missing. We can prove convergence without knowing the limit. Commented Mar 5, 2018 at 18:42
• Sometimes the practical application of a concept is in the confirmation and reinforcement of other ideas (e.g. for set theory). Cauchy sequences are ubiquitous, so it could cause a problem if we found out they had some nasty properties. But moreover, there is sometimes no prior justification for exploring a mathematical object other than intrigue...
– Jam
Commented Mar 7, 2018 at 19:28
• One might as well ask, "what is the real world application of convergence?" Have you never run an iterative computation before? Newton's method? Commented Oct 3, 2021 at 21:21
• Without an understanding of convergence, a large swathe of mathematics and mathematical physics falls to pieces. Commented Jul 25, 2022 at 9:50

As pointed out in the comments, an intuitive application of Cauchy sequences is solving an equation of the form $$f(x)=y,$$ i.e. finding a value $$x_0\in\mathbb{R}$$ such that $$f(x_0)=y$$ for some fixed $$y\in\mathbb{R}$$. The application of this is almost too vast to describe, but a simple example would be if $$x$$ is some parameter like the interest rate, $$f(x)$$ describes the bank's modeled profit depending on their interest rate, and the solution $$x_0$$ denotes the point at which they profit $$y$$.
Solving such equations are often done using variations of Newton's method, which relies on iteration to achieve an excellent approximation of the true result (up to some fixed error $$\epsilon>0$$). Proving that one can get arbitrarily close to the true result is done using: Cauchy sequences!