<s>[INST] Why is it trivial that $\left(1+\frac{2\ln3}{3}\right)^{-3/2}\leq\frac{2}{3}$? Can someone tell me why $$\left(1+\dfrac{2\ln3}{3}\right)^{-3/2}\leq\dfrac{2}{3}$$ is trivial because for me its not and I will need to do the calculation to see it.
 [/INST] We can see everything is positive, so just take reciprocal and square both sides:
$\left(1+ \frac{2 \ln 3}{3}\right) ^ {\frac{3}{2}} \ge \frac{3}{2}$
$\left(1+ \frac{2 \ln 3}{3}\right) ^ 3 \ge \frac{9}{4}$
Bring down the three inside the bracket (first step of expansion), since everything is positive the extra terms will be positive:
$1 + 2 \ln 3 + \text{stuff} \ge 1+ \frac{5}{4}$
Which is clearly true since $\ln 3 > 1$
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