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.

There are just two very small corrections that need to be made. First, there is a $k\in\Bbb N$ such that $a_k\color{red}{\le}x<a_{k+1}$. And secondly, in $(2)$ there is an $M\in\Bbb N$ such that $x+M\in\color{red}[a_\ell,a_{\ell+1})$ for some $\ell\ge N$.
