Solution
= Solution
The sample mean $S_n/n$ has expectation $p$ and variance
$$
\frac{p(1-p)}n\leq\frac1{4n}.
$$
The sum in the question is exactly
$$
\mathbb P\left(\left|\frac{S_n}{n}-p\right|>\delta\right).
$$
Chebyshev's inequality therefore gives, uniformly in $p$,
$$
\boxed{
\sum_{k\in K_\delta}\binom nkp^k(1-p)^{n-k}
\leq\frac1{4n\delta^2}}.
$$