Tchebychev's inequalities for sequences | Tangente

The Russian mathematician Pafnuty Chebyshev stands out for both the quality and the diversity of his discoveries. His research in number theory produced numerous results involving notable inequalities. One of them, known as "Chebyshev's inequality" (not to be confused with the Bienaymé–Chebyshev inequality; see the article "In probability: the Bienaymé–Chebyshev inequality"), is fairly simple to prove. Yet it is extremely useful, featuring in far more technically demanding proofs, particularly in prime number theory.

Chebyshev's inequality -------------------------

Let us consider two sequences of real numbers, both increasing or both decreasing, with exactly the same number of terms, namely n. Denote these two sequences by a₁, a₂, a₃… *a_n and b*₁, b₂, b₃… *b_n. Take, for example, the case of decreasing sequences; the reasoning is the same for increasing sequences. By assumption, a_i ≥ a_i *₊₁ and *b_i ≥ b_i *₊₁ for every integer i between 1 and n – 1.

Let us see what can be said about the product (*a_i – a_j)×(b_i – b_j), for any i and j between 1 and n. We already know that the two factors have the same sign. Since the sequences are decreasing, both factors are non-negative if i is less than j, and non-positive if i is greater than j. Their product is therefore non-negative (and may be zero, particularly when i = j). Thus, a_i b_i – a_i b_j – a_j b_i + a_j b_j* ≥ 0.

First, sum all these terms over the index i.