Prove that the set $ \displaystyle{\mathbb{N} =\{1,2,3, \cdots \} }$ is nonwhere dense in metric space $ \displaystyle{ \left( \mathbb{R} ,|\cdot| \right)}$ .
I have found a solution in two steps:
- I prove that $\displaystyle{\bar{\mathbb{N}}=\mathbb{N}}$
- $\displaystyle{ \text{int}\mathbb{N} = \emptyset }$
From these two we get that $ \displaystyle{ \text{int}\left(\bar{\mathbb{N}} \right) = \emptyset}$ so we are done.
I wonder if I we can found a solution proving that every interval $[a,b]$ has a sub-interval whose intersection with $\mathbb{N}$ is the empty set.
Any help?