Structure of dense subset of $\mathbb{R}$ Here user Mario Carneiro formulated statement which I find really interesting and useful and I proved it and I would like to know is my proof correct? 

If sequence $\{a_n\}_{n=1}^{\infty}$ implies the following conditions: 
1) $a_1<a_2<\dots<a_{n}<\dots$ 
2) $a_n\to \infty$ as $n\to \infty$
3) $a_{n+1}-a_{n}\to 0$ as $n\to \infty$
Then $\{a_n+m: (n,m)\in \mathbb{N}\times \mathbb{Z}\}$ is dense in $\mathbb{R}^1$.

Proof: Taking $x\in \mathbb{R}^1$ and let $\epsilon>0$ be given. Then $\exists k\in \mathbb{N}$ such that $a_k<x<a_{k+1}$ (assuming that $x>a_1$). Then $\exists N:=N_{\epsilon}$ such that $n\geqslant N$ implies $a_{n+1}-a_n<\epsilon$.
1) If $k\geqslant N$ then $|x-a_k|<\epsilon.$
2) If $k<N$ then $\exists M\in \mathbb{N}$ such that $x+M\in(a_l, a_{l+1})$ where $l\geqslant N$ then $|x+M-a_l|<\epsilon.$
If $x\leqslant a_1$ then $\exists K\in \mathbb{N}$ such that $x+K>a_1$ and we can apply previous reasoning for $x+K$.
Q.E.D.
Remark: If $A$ is dense subset of $\mathbb{R}^1$ then: 1) for any $r>0$ the set $rA$ also dense subset of $\mathbb{R}$. 2) for any $c\in \mathbb{R}$ the set $c+A$ also dense.

Since the original proof I had in mind differs a little, I'll write it down here. (Also it turns out that the monotonicity assumption $a_1\le a_2\le\dots$ is not necessary.)
Take $\epsilon>0$. By assumption (2), there is some $N$ such that $a_{n+1}-a_n\le\epsilon$ for all $n\ge N$.
Lemma: The sequence $a_n$ is $\epsilon$-dense in $[a_N,\infty)$. Proof: Take $x\ge a_N$ and let $k\ge N$ be the least integer such that $a_k\ge x$. Existence is guaranteed by assumption (3). Then $a_k-\epsilon\le a_{k-1}<x$ since $a_{k+1}-a_k\le\epsilon$ and $\lnot(a_{k-1}\ge x)$, so $a_k\in[x,x+\epsilon)$ and thus $a_k$ is $\epsilon$-close to $x$.
Returning to the main theorem, given $x\in\Bbb R$ we can find an $m\in\Bbb Z$ such that $x-a_N-m\in[0,1)$ (explicitly, $m=\lfloor x-a_n\rfloor$). Then by the lemma there is an $a_k$ which is $\epsilon$-close to $x-m$, and so $a_k+m$ is $\epsilon$-close to $x$. Thus $\{a_n\}_{n=1}^\infty+\Bbb Z$ is $\epsilon$-dense in $\Bbb R$ for all $\epsilon$, which is equivalent to it being dense in $\Bbb R$.
