The Cauchy convergence criterion

The Cauchy condition states that for every positive tolerance $\varepsilon$, there exists an index $N$ such that for all indices $m, n > N$, the distance between the sequence terms $|a_m - a_n|$ is strictly less than $\varepsilon$. This means that all sufficiently late terms in the sequence become arbitrarily close to each other, regardless of how far apart their indices are.

Conditions: The sequence is defined over the real numbers.; $\varepsilon > 0$ is an arbitrary positive tolerance.; $N$ is an integer threshold that depends on $\varepsilon$.; $m$ and $n$ are any indices strictly greater than $N$.

The standard $\varepsilon$-$N$ definition of convergence requires the existence of a specific limit $L$ such that $|a_n - L| < \varepsilon$ for all $n > N$. In contrast, the Cauchy condition does not reference any external limit; it only requires that the terms get close to *each other*, i.e., $|a_m - a_n| < \varepsilon$ for all $m, n > N$.

Conditions: Comparing the two definitions for sequences in a metric space.; Standard convergence: $\exists L, \forall \varepsilon > 0, \exists N, \forall n > N: |a_n - L| < \varepsilon$.; Cauchy condition: $\forall \varepsilon > 0, \exists N, \forall m, n > N: |a_m - a_n| < \varepsilon$.

In the visualization of the Cauchy condition, the horizontal band represents the tolerance $\varepsilon$, defining the maximum allowed distance between any two terms in the tail. The vertical cutoff line represents the index $N$, marking the point beyond which all sequence terms must lie within the band relative to each other.

Conditions: The visualization plots sequence index on the horizontal axis and value on the vertical axis.; The band is centered around the cluster of tail terms.; The cutoff is a specific index $N$.

Testing only consecutive terms is insufficient because the Cauchy condition requires that *every* pair of sufficiently late terms is close, not just adjacent ones. A sequence can have consecutive differences approaching zero while still diverging or failing to cluster tightly over long intervals.

Conditions: The sequence is defined over the real numbers.; The condition checks $|a_{n+1} - a_n| \to 0$.; The Cauchy condition requires $\forall \varepsilon > 0, \exists N, \forall m, n > N: |a_m - a_n| < \varepsilon$.

A finite animation can only display a limited number of terms and a specific tolerance $\varepsilon$. The Cauchy condition, however, is a universal statement: it must hold for *every* positive $\varepsilon$ and *every* pair of indices beyond some $N$.

Conditions: The animation shows a finite number of sequence terms.; The animation uses a specific, fixed tolerance $\varepsilon$.; The Cauchy condition requires $\forall \varepsilon > 0$ and $\forall m, n > N$.

The completeness of the real numbers guarantees that every Cauchy sequence of real numbers converges to a real limit. In other words, the Cauchy condition is equivalent to convergence in $\mathbb{R}$.

Conditions: The sequence consists of real numbers.; The sequence satisfies the Cauchy condition.; The underlying space is the set of real numbers $\mathbb{R}$.