With this equivalence relation on R:
$x \sim y \Leftrightarrow x = y \lor x,y \in \mathbb{Z}$
I have to proof that the topologic quotient $X/\sim$ is Frechet Urysohn, which means that for evert $A \subset X/\sim$ and $p \in Cl(A)$, there is a sequence of elements of A such that $(a_n) \Rightarrow p$.
First, I studied $X/\sim$ and I can show that it contains an equivalence class for $\mathbb{Z}$ and the others look like $[x] = \{x\}$ for $x \in \mathbb{R} - \mathbb{Z}$. Now, about the proof, I have seen other answers in this forum that differentiate when $p \in A$ or p is a limit point of A, but I have been asked to do it using the definition $p \in Cl(A) \Leftrightarrow \forall U$ neighborhood of p $U \cap A \not = \emptyset$. Up until now, I chose to differentiate two situations: $p \in \mathbb{Z}$ or $p \not \in \mathbb{Z}$. For the second case, I can see easily that $p \in (a, a+1)$ for some $a \in \mathbb{Z}$ and that $(a, a+1) \cap A \not = \emptyset$. Is it enough to take the elements in $(a, a+1) \cap A$ and construct a sequence with them. How can I define it? For the first case, I am actually a little confused because I am not sure how would be A in the case that $p \not \in A$ due to the fact that it could be the limit point of other subset and I guess it is not possible to characterize $A$ to a general case. If you could help to see what I'm missing in this proof, I would really appreciate it.
HINT: Let $X=\Bbb R/\!\sim$, let $q:\Bbb R\to X$ be the quotient map, and let $z=q(0)$ (so that $q[\Bbb Z]=\{z\}$. Let $E$ be the set of functions $\epsilon:\Bbb Z\to\left(0,\frac12\right)$, and for each $\epsilon\in E$ let $U_\epsilon=\bigcup_{n\in\Bbb Z}\big(n-\epsilon(n),n+\epsilon(n)\big)$; it’s not too hard to show that $\{q[U_\epsilon]:\epsilon\in E\}$ is a local base of open nbhds of $z$ in $X$. Use this to show that if $z\in\operatorname{cl}_XA$, there must be an $n\in\Bbb Z$ such that
$$n\in\operatorname{cl}_{\Bbb R}\big(q^{-1}[A]\cap(n-1,n+1)\big)\,.$$
This is of course trivial if $z\in A$, so you can concentrate on the case $z\in\operatorname{cl}_XA\setminus A$.