Solution
= Solution
The preceding representation and the <central limit theorem> give
$$
\frac{B_n-np}{\sqrt{npq}}
\ \xrightarrow{\ d\ }\ N(0,1).
$$
Since
$$
S_n-n(p-q)=2(B_n-np),
$$
one may take
$$
\boxed{b_n=n(p-q),\qquad c_n=2\sqrt{npq}}.
$$
For $0<p<1$ this gives the required convergence to $\Phi(x)$ at every real $x$.