Remember \(x^2 \geq 0\)
But in general, even powers are positive, which can help you prove something is \(\geq 0.\)
About Euler’s Constant …
Certainly \(\lim_{n \to \infty} \bigl(1 + \frac{1}{n}\bigr)^n = \mathrm{e},\) but to keep with the theme, for all positive integers \(n,\) \[\biggr(1 + \frac{1}{n}\biggl)^n \lt \biggr(1 + \frac{1}{n+1}\biggl)^{n+1} \lt \mathrm{e}.\]
Cauchy-Schwarz Inequality
For any collection of real numbers \((a_1, a_2, \dotsc, a_n)\) and \((b_1, b_2, \dotsc, b_n),\) \[ \Biggl(\sum_{i=1}^{n}a_i b_i\Biggr)^2 \leq \Biggl(\sum_{i=1}^{n}a_i\Biggr)\Biggl(\sum_{i=1}^{n}b_i\Biggr). \] Phrased differently, if you’ve ever seen an inner product space before, \[ |\langle x,y \rangle| \leq \|x\|\cdot\|y\|\,. \]
Arithmetic Mean, Geometric Mean (AM/GM) Inequality
For nonnegative real numbers \(a_1, a_2, \dotsc, a_n,\) \[ \sqrt[n]{a_1a_2\dotsb a_n} \leq \frac{a_1+a_2+\dotsb+a_n}{n}\,, \] where equality holds only when all the \(a_i\) are equal. The expression on the left-hand side of the inequality is called the geometric mean of those numbers, and the expression on the right-hand side is called the arithmetic mean. Occasionally this inequality will be extended to include another expression, the harmonic mean, too, \[ \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_n}} \leq \sqrt[n]{a_1a_2\dotsb a_n} \leq \frac{a_1+a_2+\dotsb+a_n}{n}\,. \] which altogether constitute the three Pythagorean means of antiquity.