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A sequence $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence if for every positive $\varepsilon$, there exists a natural number $N$ such that for all indices $n,m\ge N$, the distance between the corresponding terms satisfies $|a_n-a_m|<\varepsilon$.

Conditions: The sequence consists of real numbers.; The condition must hold for every $\varepsilon>0$.; The threshold $N$ may depend on $\varepsilon$.; Both indices $n$ and $m$ must be greater than or equal to $N$.

A sequence $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence if for every positive $\varepsilon$, there exists a natural number $N$ such that for all indices $n,m\ge N$, the distance between the corresponding terms satisfies $|a_n-a_m|<\varepsilon$.

Conditions: The sequence consists of real numbers.; The condition must hold for every $\varepsilon>0$.; The threshold $N$ may depend on $\varepsilon$.; Both indices $n$ and $m$ must be greater than or equal to $N$.