If we have two binomials $B(n,p_1)$ and $B(n,p_2)$, what statements can we make about the similarity between them as $\mid p_1 - p_2\mid \rightarrow 0$? If $p_1$ were constant and $p_2$ monotonically approaches it, does $\mid B(k,n,p_1) - B(k,n,p_2)\mid \rightarrow 0$ monotonically for any $k$ number of successes?
Either way, are there any theorems we can appeal to to give answers, or does this have to be answered analytically? Concepts like convergence in probability and distribution don't seem to be appropriate because they rely on limits of the random variables?