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Suppose that you have a convergent sequence of real numbers $x_n$ and assume that you want to impose conditions so that this sequence converges to some limit $x^*$ for $n\rightarrow \infty$ in a smooth way that is without jumps. In other terms I want to impose that it does not converge too fast and that it doesn't converge to $x^*$ abruptly at some point in time. In other terms, I want the function $|x_n - x^*|$ to be continuous and with the x-axis as a horizontal asymptote.

If I just imposed the sequence to be a Cauchy sequence, I'd not get the result since a constant sequence is Cauchy. I'd modify the Cauchy property this way:

for every positive real number $\epsilon$ there is a positive integer $N$ such that for all natural numbers $n,m>N$ we have $|x_n - x_m|<\epsilon$ and $|x_n - x_m|=\epsilon$ for at least one pair $m,n$

Is this correct? Is there a more elegant way to get the result?

user279687
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    You will have to say something like for all rational $\epsilon$. If not, there are uncountably many choices for $\epsilon$, but only countably many values of the form $|x_n-x_m|$. – JLinsta Mar 06 '21 at 15:42

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The problem with this idea is that sequences are discrete objects, so it doesn't make much sense to discuss their "continuity". In fact, any sequence viewed as a map $\mathbb{N} \to \mathbb{R}$ is continuous. The best you can probably say is to impose the restriction, $$|x_1 - x^*| > |x_2 - x^*| > |x_3 - x^*| > \cdots > 0$$ and this sequence $|x_n - x^*| \searrow 0$.

JLinsta
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