Prove that a power series that is zero on a sequence that converges to zero is the zero function
Because we have convergence on a interval around zero, convergence is uniform for both the series and the series of derivatives, and so we can differentiate term by term.
You have
$$
b_1=g'(0)=\lim_{h\to0}\frac{g(h)-g(0)}{h}=\lim_k\frac{g(x_k)-g(0)}{x_k}=0.
$$ We now have that $$g(x)=x\sum_{n=2}^\infty b_n x^{n-1}.$$ Since $x_n\ne0$ and $g(x_n)=0$, we get that
$$
g_2(x_k)=\sum_{n=2}b_n(x_k)^{n-1}=0
$$
for all $k$. Since $g_2(0)=0$, we may apply the above reasoning again, to obtain $b_2=0$. Now we can repeat the argument to obtain $b_3=0$, $b_4=0$, etc.