Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Prove Inequalities 1 and 2 in Table 3.2.1.
Proof of Inequality 1: 0≤(|x|−y)2 = x2+y2−2x y ⇒ x y ≤x2+y2/2.
Proof of Inequality 2: 0≤(x+|y|)2 = x2+y2+2x y ≤x2+y2+2x2+y2/2=2x2+y2.
Since (x+|y|)2≤2x2+y2, it follows that x+|y|≤2 x2+y2.
(Note the use of Inequality 1 in the proof of Inequality 2.)
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