Show that if $NP\subseteq BPP$ then also $NP=RP$ - considerations about solution
It's true that we can run $A$ even more times, but $O(\log n)$ times suffices for us to obtain a certainty of $1 - 1/(100n)$ (say) for each $x_i$. Applying the union bound, we deduce that the algorithm finds a valid truth assignment with probability at least $1 - 1/100$.

If you run $A$ polynomially many times then you get an exponentially small error, which is not needed here.