I have another question about proving "For every real number $x$, there's exactly one integer $n$ such that $n \leq x\lt n+1$".
Let $A=\left \{ n \in \mathbb{Z} \mid n \leq x \right \}$. Let $\hspace{2mm} B=\left \{ n \in \mathbb{Z} \mid x < n+1 \right \}$
Now, how do I know $A\cap B \neq \varnothing$ ?
Playing around with "$\mathbb{Z}$ is not bounded" only gave me that $A$ and $B$ exists, but I don't see how I can get $A\cap B \neq \varnothing$
Thanks in advance.