If $A\subset B$ and $B=V_1\cup V_2$, where $V_1$ and $V_2$ are open sets, why are the sets $A\cap V_1$ and $A\cap V_2$ open?

Question

In a proof of a question I found the following sentence:

"Since A⊂B, it follows that A∩V1 and A∩V2 are open in (A,d)."

Why does it hold?

I understand that if $x\in A\cap V_1\Rightarrow x\in A \wedge x\in V_1$. Now, $V_1$ is an open set, thus $\forall x\in V_1$ we can find an open ball $B(x,\epsilon)\subset V_1$.

Does it guarantee that $B(x,\epsilon)\subset A∩V_1$ (so that $A∩V_1$ to be open)?

I'm kind of confused. I think that i'm missing something. Can someone help me please?