If a1 ≥ a2 ≥ ... ≥ an and b1 ≥ b2 ≥ ... ≥ bn then
|akbk ≥||(|| n
The proof is that the RHS can be written as the sum of n different
(a1b1 + a2b2 + ... + an-1bn-1 + anbn) +
(a1b2 + a2b3 + ... + an-1bn + anb1) +
(a1bn + a2b1 + ... + an-1bn-2 + anbn-1)
The first of these n sums is at least as big as each of the others by the rearrangement inequality, and n times this first sum is the LHS, so the LHS is at least as big as the RHS.
The AM-GM Inequality: the Arithmetic Mean of positive numbers is always greater than the Geometric Mean. This is proved using Jensen's Inequality.
The Cauchy-Schwarz inequality
The Triangle Inequality
The Quadratic Formula
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