How could I prove that $B$ reduces to $A$ in polynomial time in this case?
Take $a\in A$ and  $b \notin A$.

the reduction defined as follows:
$$f(x)=\begin{cases}
a &amp; x\in B\\
b &amp; x\notin B
\end{cases}$$

It is easy to see that $f \in \text{POLY}$ since $B\in\text{P}$, and the correctness is trivial.