let X be a metric space and A an open subset of X

Prove that A ∩B^{−}⊂ (A∩B)^{−}

My approach:

let x∈A ∩B^{− }Then x∈A and x∈B^{−}

nbhd(x) intersects A and B for all ε >0

so x∈A^{−}∩B^{−}

but (A∩B)^{−}⊂A^{−}∩B^{−}

Did I do anything wrong or miss something?