Non-Numerical proof of $e<\pi$

Solution 1:

Just a suggestion.

$$\int_{-\infty}^\infty\frac{\cos x}{x^2+1}\operatorname d\!x=\frac\pi e$$

If you can prove that the above integral is $>1$ you know $\pi > e$.

Solution 2:

If we define $e$ as $\lim_{n\to\infty}\left(1+\frac1n\right)^n$, then $e<3$ because, for each $n\in\mathbb{N}$,\begin{align}\left(1+\frac1n\right)^n&\leqslant1+1+\frac1{2!}+\cdots+\frac1{n!}\\&<1+1+\frac12+\frac1{2^2}+\cdots+\frac1{2^{n-1}}\\&<3.\end{align}On the other hand, $2\pi$ is greater than the perimeter of a regular hexagon inscribed in a circle with radius $1$, which is $6$. Therefore, $\pi>3>e$.