# Equivalence classes and sequences

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 $$a_n-f_n(a,a_n,"\mathcal{R}")\xrightarrow[n]{}0$$?

• How do you define $a$ mod 10 for real numbers ?
– Ulli
Commented Jun 18 at 14:22
• 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.
– VAL
Commented Jun 18 at 16:55

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.