In something I am reading, the following statement is mentioned in passing as something obvious: if $X_1,\ldots,X_n$ are i.i.d. Bernoulli with parameter $1/2 + \delta$, then $\mathbb{P}(\sum_{i=1}^n X_n < n/2) \ge \exp(-n\delta^2)$. However, I am not sure how to show this. [The inequality may also be ignoring constants.]
There is a lower bound in terms of entropy on Wikipedia, but I am wondering if there is a simpler approach that gives the weaker result above.