= Solution
Suppose for a contradiction that $A\cap B=\varnothing$. Each grid point then belongs to exactly one of $A$ and $B$; colour it red in $A$ and green in $B$. By part (a), there are adjacent points $a_n\in A$ and $b_n\in B$, in either order, with
$$
|a_n-b_n|=\frac1n.
$$
The interval $I$ is a <compact set>, so a <subsequence> of $(a_n)$ converges to some $x\in I$. Along the same subsequence $b_n\to x$. Since $A$ and $B$ are <closed sets>, $x\in A\cap B$, a contradiction. Hence $A\cap B\ne\varnothing$.
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