I have the following summation:
$$\beta = \sum\limits_{n=0}^{\infty}\sum\limits_{j=0}^{[\frac{n}{k}]} P(N_0=j)P(N_1=n-j)$$
Here, $N_0$ is negative binomial (number of tails required to see a total of $m$ heads when tossing a coin with probability of heads, $p$) and $N_1$ is negative binomial with parameters $m$ and $p-\delta p$ and $k \in (2,3, \dots n-1)$. So,
$$P(N_0=j) = {j+m-1\choose j}p^m(1-p)^j$$ and $$P(N_1=n-j) = {n-j+m-1\choose j}(p-\delta p)^m(1-p+\delta p)^{n-j}$$
I need to prove:
$$\lim_{m \to \infty} \beta = 0$$
I have verified this through explicitly calculating the summation for various values of $p$, $\delta p$. For example:
