Proposition 4. Assuming that:

$0 < 𝜀 < \frac{1}{4}$

Then

\sum_{i = 0}^{⌊ (\frac{1}{2} - 𝜀) n ⌋} (\binom{n}{i}) \leq \frac{1}{𝜀} e^{- \frac{𝜀^{2} n}{2}} \cdot 2^{n} .