Proving inequalities means to establish that theinequality holds true for arbitrary admissible valuesof the parameters.Example 1:Prove that the absolute value of asum does not exceed the sum of the absolute values:|a+b|≤|a|+|b|.①Proof.The absolute value of the sum |a+b| isequal to a+b or to -(a+b).From the definition ofthe absolute value we havea≤|a|,b≤|b|and combining these inequalities termwise,we geta+b≤|a|+|b|.②In exactly the same manner,-a≤|a|,-b<|b| and-(a+b)≤|a|+|b|.③From the inequalities ②,③ and the definition of the Proving ine the absolute value of asum does not exceed the sum of the absolute values: | a + b | ≦ | a | + | b |. A) the absolute value of the sum | a + b | isequal to a + b or to - (a + b) .From the definition of the absolute value we havea≤ | a |, b≤ | b | and combining these inequalities termwise , we geta + b ≦ | a | + | b |. ② In exactly the same manner, -a ≦ | a |, -b <| b | and- (a + b) ≦ | a | + | b | the inequalities ②, ③ and the definition of the