$\textbf{Cauchy sequence}$
A sequence of real numbers $(a_n)_{n\in\N}$ is called a Cauchy sequence if for every real number $\varepsilon>0$ there exists $N\in\N$ such that $$\forall m,n\geq N:\space\lvert a_n-a_m\rvert<\varepsilon.$$
$\textbf{Cauchy sequence}$
A sequence of real numbers $(a_n)_{n\in\N}$ is called a Cauchy sequence if for every real number $\varepsilon>0$ there exists $N\in\N$ such that $$\forall m,n\geq N:\space\lvert a_n-a_m\rvert<\varepsilon.$$
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𝐂𝐚𝐮𝐜𝐡𝐲 𝐬𝐞𝐪𝐮𝐞𝐧𝐜𝐞
A sequence of real numbers (𝑎ₙ)ₙ∈ℕ is called a Cauchy sequence if for every real number 𝜀>0 there exists 𝑁∈ℕ such that
∀𝑚,𝑛≥𝑁: ∣𝑎ₙ−𝑎ₘ∣<𝜀.