I have to prove the following inequality: $e^{-2} < \ln2.$
Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper estimate for $e^{-2}$ and a lower estimate for $\ln2$. Using this method, I got that $e^{-2} \leq \frac{1}{4}$, but I couldn't proceed from that point.
Any hints would be much appreciated.