Suppose that $(x_i)_{i \geq 1}$ is not a Cauchy sequence of real numbers.
How to prove that there exist $\varepsilon >0 $ and an increasing sequence $(i_n)$ of indices such that $$ |x_{i_{n+1}}-x_{i_n}|\geq \varepsilon $$ for any $n \geq 1$?
2026-03-27 07:11:39.1774595499
Negation - Cauchy sequence
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