$\displaystyle Let \ x \in R \ with \ x>0 \ . show \ that \ there \ is \ n \in N such \ that \ n-1 \leq x < n . $
Oops! How did I do that mistake
Then, let me correct.
Suppose that for every $\displaystyle n\in\mathbb{N}$, $\displaystyle x\not\in[n-1,n)$. However, since $\displaystyle x$ is fixed, we can find $\displaystyle m\in\mathbb{N}$ such that $\displaystyle x<m$, i.e., $\displaystyle x\in[0,m)$
Then, we know that $\displaystyle x\not\in[m-1,m)$, similarly $\displaystyle x\not\in[m-2,m-1)$, and by following similar arguments, we show that $\displaystyle x\not\in[m-m,m-m+1)=[0,1)$.
This shows that $\displaystyle x$ must be negative, and thus is a contradiction.
But I have to say that this is not a nice proof.
Thanks to Plato for pointing out my mistake.