Q:

Proving: If $|x-y|<\frac{1}{n}$ for every natural $n$ then $x=y$

It's really basic but I am trying to prove:
$$
\forall x,y\in \mathbb{R}.(\forall n\in \mathbb{N}. |x-y| <\frac{1}{n} \Rightarrow 
x=y)$$
I tried to prove it by proving that that if $$A= \left\{ \frac{1}{n} \ \middle| \ n\in\mathbb{N} \right\}$$
then the $$\operatorname{inf}(A) =0$$ but I couldn't find a way to use this after I proved it.
Unfortunately, I am not allowed to use limits here. 

A:

Suppose $x\neq y$, then
$$\vert{x-y}\vert>0$$
by the Archimedian property of $\mathbb{R}$ there exists $n\in\mathbb{N}$ such that:
$$n\vert{x-y}\vert>z,\quad \forall z\in\mathbb{R}\ \text{with}\ z>0$$
take in particular $z=1$, 
so there exists $m\in\mathbb{N}$ such that:
$$m\vert{x-y}\vert>1$$
which gives
$$\vert{x-y}\vert>\frac{1}{m}$$
this is a contradiction.

