Natural Number Inductive Proof

Question

Show that $[1,\infty)=\bigcup_{n∈N}[n,n+1)$ and that $[n,n+1)∩[m,m+1)\neq ∅$ only if $n=m$.

To prove $[n,n+1)∩[m,m+1)\neq ∅$ only if n=m I think I need to use contradiction. But I have little idea how to start this proof.

Answer 1

Assume r is in the intersection.
There are three cases: n < m, n = m and n > m.
Show the first and third cases are contradictions.

Answer 2

Suppose $n,m\in \Bbb N$ with $n<m.$ Then $n+1\le m.$ So for any $x\in [n,n+1)$ and for any $y\in [m,m+1)$ we have $x<n+1\le m\le y,$ which implies $x<y,$ which implies $x\ne y.$ So $[n,n+1)$ and $[m,m+1)$ have no members in common.