Theorem
a,b∈Ratn
n∈Nat
(a∙b)2≤Q(a)Q(b)
Proof
Recall that by definition
a∙b=n∑i=1aibiQ(a)=n∑i=1a2iQ(b)=n∑i=1b2i.
The claim, reformulated as
(n∑i=1aibi)2≤(n∑i=1a2i)(n∑i=1b2i),
is equivalent to (by distributing terms)
(n∑i=1a2ib2i)+∑1≤i<j≤n2aibiajbj≤(n∑i=1a2ib2i)+∑1≤i<j≤na2ib2j+a2jb2i
and (by collecting terms)
0≤∑1≤i<j≤na2ib2j+a2jb2i−2aibiajbj=∑1≤i<j≤n(aibj−ajbi)2.
The last statement is true because squares are positive and sums of positive numbers are positive.
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