The most usual definition for a Cauchy sequence use the infinitesimal variable $\varepsilon$. But I found only here the following definition: $\lim_{\min(m,n)\to\infty}d(a_m,a_n)=0$, where $d$ is the distance. This definition is much more intuitive for me, but I would like to know if it is correct. Thanks in advance.
2026-04-04 01:18:25.1775265505
An alternate definition for Cauchy Sequence
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Saying that distance between a and b limits to zero is the same as saying that absolute value of difference of a and b is smaller than some positive number epsilon, because your can choose ANY epsilon you like. You can choose epsilon that is "almost" zero (really small). In other words the distance between a and b in that case is "almost" 0.
Geometric interpretation of absolute value of the distance between a and b on real line is actually exactly the distance between a and b.