Let $X= B(n,1/2)$, $Y=B(n,1/2 + \delta)$, for a small $\delta >0$ be two Binomial Distributions.
I am looking for a lower bound on the Total Variation Distance the two Binomials $X,Y$.
My attempt at deriving a lower bound is the following: Since $X, Y$ have huge variance we can approximate each of them very well with a discretized Normal and then lower bound the total variation distance of the two Normals. My problem here is that I am not sure how to go from discretized Normals to continuous Normals.
Having two discretized normals as defined in this paper which are in Total Variation distance $\epsilon$ then is it true that the continuous Normals with the same mean and variance are also in total variation distance at most $\epsilon$ ?