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Math Help - analysis of a proof

  1. #16
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    Quote Originally Posted by xalk View Post
    He is a bit confused but the problem is quite complicated
    That is true only for people who do not understand the question.
    Are you among them?
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  2. #17
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    Quote Originally Posted by Plato View Post
    That is true only for people who do not understand the question.
    Are you among them?
    I think considering this is the university help section, it is fair to say this proof is not complicated. It just relies on the properties of the real numbers.

    x\leq |x| follows from the fact that the reals are ordered.
    a \in \mathbb{R}, b \in \mathbb{R} \implies ab \in \mathbb{R} depends on how you look at R, easiest reasoning is because R is a field.

    The rest follows from simple algebra.
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  3. #18
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    Quote Originally Posted by Plato View Post
    That is true only for people who do not understand the question.
    Are you among them?
    The statements of the proof are :

    1) ab\leq |ab|

    2) 2ab\leq 2|a||b|

    3)  (a+b)^2\leq (|a|+|b|)^2

    4)  |a+b|\leq |a|+|b|

    And the problem ask us :

    a) To mention appropriate theorems or definitions that are responsible for those statements.

    b) How are they involved .

    For the (a) part the following tabular form shows clearly the theorems or definitions involved in the proof.


    1) ab\leq |ab|.................................................. ............by using the theorem : for all ,x: x\leq |x|

    2)|ab| = |a||b|............................................ ..................................by using the theorem : for all , x, y : |xy| = |x||y|

    3)  ab\leq |a||b|.................................................. .........by substituting (2) into (1)


    4) 2ab\leq 2|a||b|.................................................. .......by using the theorem : for all ,x,y,z : z>0 and x\leq y\Longrightarrow xz\leq yz

    5) a^2+b^2+2ab\leq 2|a||b|+a^2+b^2.............................by using the theorem : for all ,x,y,z :  x\leq y\Longleftrightarrow x+z\leq y+z


    6) a^2 = |a|^2.................................................. ............by using the theorem :for all ,x : x^2 = |x|^2


    7) b^2 = |b|^2.................................................. ............by using again the same theorem as in (6)

    8)  a^2+b^2+2ab\leq 2|a||b|+|a|^2+|b|^2.................................................. ..........by substituting (7) and (6) into (5)

    9)  (a+b)^2 = a^2+b^2+2ab............................................by using the identity (theorem) : for all , x,y : (x+y)^2 = x^2+y^2+2xy


    10) (|a|+|b|)^2 = |a|^2+|b|^2+2|a||b|...........................by using the same theorem as in (9)


    11) (a+b)^2\leq (|a|+|b|)^2............................................by substituting (10) and (9) into (8)


    12) \sqrt {(a+b)^2}\leq\sqrt {(|a|+|b|)^2}........................by using the theorem : for all ,x,y :  x\geq 0 and  y\geq 0\Longrightarrow (x\leq y\Longleftrightarrow\sqrt x\leq\sqrt y)



    13) \sqrt{(a+b)^2} = |a+b|..............................................by using the theorem : for all ,x : \sqrt{ x^2} = |x|


    14) \sqrt{(|a|+|b|)^2} = ||a|+|b||.............................................by using the same theorem as in (13)


    15)  |a+b|\leq ||a|+|b||.................................................. ..........by substituting (14) and (13) into (12)


    16) ||a|+|b|| = |a|+|b|........................................... .........................by using the definition of absolute value : for all , x :  x\geq 0\Longrightarrow |x| = x


    17)  |a+b|\leq |a|+|b|.................................................. .by substituting (16) into (15).


    Note: that in the above many other statements that are not mention in the proof are exposed . To include them all or part of them in the proof it is matter that depends on the writer's style .


    NOW for part (b) .

    Part (b) deals with the laws of logic ,the application of which, transforms the theorems involved in the proof into the statements of the proof.

    That part i leave to Plato ,since he thinks that the whole problem is not complicated.

    Of course if he wishes to do that
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