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Suppose I'm working in $\mathbb{R}$ and I have the equivalence relation, $a\mathcal{R} b \leftrightarrow a \text{ mod } 10=b \text{ mod } 10$. Suppose now that I'm given a sequence $\{a_n\}_n\in \mathbb{R}$.

Take the quotient space $\mathbb{R}/ \sim$ and suppose that $[a_n]\xrightarrow[n]{} [a]$ for some $a\in \mathbb{R}$. what can I say about $\{a_n\}_n$?

My guess is that $\forall n\exists k_n\in \mathbb{N}$: $a_n-10 k_n-a\xrightarrow[n]{} 0$ and I can't say much more than this.

Is this true? Are there some known results in a more general setting(I guess that there are but I don't know what to look for)? That is, suppose that I have $\{a_n\}_n\in \mathcal{H}$, $\mathcal{R}$ is some equivalence relation and $[a_n]\xrightarrow[n]{} [a]$ for some $a\in \mathbb{R}$. Can I always find a family of functions $\{f_n\}_n$(in some functional space) such that $a_n-f_n(a,a_n,"\mathcal{R}")\xrightarrow[n]{}0$?

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    $\begingroup$ How do you define $a$ mod 10 for real numbers ? $\endgroup$
    – Ulli
    Commented Jun 18 at 14:22
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    $\begingroup$ You extend the definition of the one of natual numbers, if you are familiar with programming it is the % operator: I think that you can use this definition a mod b=c iff a-c=nb for some $n \in \mathbb{N}$. For example: 12.5 mod 10=2.5, -30.6 mod 6=-0.6. Basically it is the remainder of a/b. $\endgroup$
    – VAL
    Commented Jun 18 at 16:55

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Given the sequence $\{a_n\}_n\subseteq\mathbb{R}$ and the equivalence relation $aRb\Longleftrightarrow a\mod10=b \mod 10$, we consider the quotient space $\mathbb{R} / \sim _.$ If $[ a_n] \to [ a]$ as $n\to\infty$ for some $a\in\mathbb{R}$,

we aim to understand the behavior of the sequence $\{a_n\}_n.$

First, observe that since $a_n$ mod $10\to a$ mod 10,for every $n$,there exists an integer $k_n$ such that $$a_n=10k_n+(a\mod10).$$ Given that $a_n\to a$,we can write $$a_n-10k_n\to a-(a\mod10)\mathrm{~as~}n\to\infty.$$

Since $a-(a$ mod 10) is simply $10\left\lfloor\frac a{10}\right\rfloor$,we have $$a_n-10k_n\to10\left\lfloor\frac{a}{10}\right\rfloor \quad \text{as} \quad n\to\infty.$$ Thus, if $[a_n]\to[a]$ in the quotient space, it follows that $$\forall n,\exists k_n\in\mathbb{Z}\text{ such that }a_n-10k_n\to10\left\lfloor\frac{a}{10}\right\rfloor.$$

A general example: suppose we have a sequence $\{a_n\}_n\subseteq\mathcal{H}$,where $\mathcal{H}$ is a topological space and $R$ is some equivalence relation. If $[a_n]\to[a]$ for some $a\in\mathcal{H}$, the behavior of $\{a_n\}_n$ can often be described using a family of functions $\{f_n\}_n$ such that $$a_n-f_n(a,a_n,\mathbb{R})\to0\text{ as }n\to\infty.$$ Obviously, we find $f_n$ ensuring the convergence.

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