Prove a $\pi$ inequality: $\left(1+\frac1\pi\right)^{\pi+1}<\pi$
Solution 1:
$$g(x)=\left(1+\frac{1}{x}\right)^{x+1}$$ is a decreasing function over $\mathbb{R}^+$ in virtue of the Bernoulli inequality.
Since $\frac{22}{7}>\pi$ (the Archimedean approximation), it is sufficient to show that $g(22/7)<\pi$.
Since $\left(1+\frac{7}{22}\right)^{29} < 3015$, it is sufficient to show that $\pi^7>3015$. This follows from $\pi>3.141$.