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Let $(x_n)_{n \in \mathbb{N}}$ a real convergent sequence to a number $l$. We can prove that $(\left| x_n - l\right|)_{n \in \mathbb{N}}$ is ultimately monotone?

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No, we cannot. For example, let $x_n=1/n$ if $n$ is odd, and $2/n$ if $n$ is even.

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No, the sequence $$ \frac13, \frac12, \frac15, \frac14, \frac17, \frac16, \ldots $$ converges to $0$ but is never monotonic.

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