In this paper by Winter et al, it is stated that the following 'Chernoff bound' holds for $i.i.d.$ random variables $X_k$ such that $0 \leq X_k \leq 1$ and $E(X_k) = p$ for all $k$,
$$\Pr \left[ \frac{1}{N}\sum_{k=1}^N X_k \geq (1+\eta)p \right]\leq \exp(-\frac{N\eta^2p}{2\ln 2}),$$ and $$\Pr \left[ \frac{1}{N}\sum_{k=1}^N X_k \leq (1-\eta)p \right]\leq \exp(-\frac{N\eta^2p}{2\ln 2}),$$ even without a condition on the varibale $\eta$. However, this $2\ln2$ exponent is the smallest among the variants of the Chernoff bounds (usually it's $2$ or $3$) I've ever seen, and I can't find a proof for this version. Is this Chernoff bound even true? How can one prove this version?