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Proof of Cauchy's Inequality ... Sohrab Proposition 2.1.23

  1. Aug 7, 2017 #1
    1. The problem statement, all variables and given/known data

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

    I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

    I need help with Proposition 2.1.23/Exercise 2.1.24 (Exercise 2.1.24 asks readers to prove Proposition 2.1.23)

    Proposition 2.1.23/Exercise 2.1.24 reads as follows:

    ?temp_hash=4a50669fed0bf5cd884e74242a2c91d0.png

    In the above text by Sohrab, we read the following:

    " ... ... Observe that for any ##t \in \mathbb{R}, F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0## and look at the discriminant ##Z^2 - XY## of ##F(t)##. ... ... "


    Can someone please explain how one proceeds after determining the discriminant and what principles/properties of the discriminant are used ...

    2. Relevant equations

    The relevant information is given in the problem statement ... basic properties of the real numbers are assumed ...

    3. The attempt at a solution

    my working so far on the exercise is as follows:

    ##F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0##

    Now ... ... ##\sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0##

    ##\Longleftrightarrow \sum_{ i = 1 }^n ( x_i^2 - 2t x_i y_i + t^2 y_i^2) \ge 0##

    ##\Longleftrightarrow \sum_{ i = 1 }^n x_i^2 - 2 t \sum_{ i = 1 }^n x_i y_i + t^2 \sum_{ i = 1 }^n y_i^2 \ge 0## ... ... ... (1)


    ... now let ##X = \sum_{ i = 1 }^n x_i^2##, ## \ ## ##Y = \sum_{ i = 1 }^n y_i^2 \ ## and ## \ Z = \sum_{ i = 1 }^n x_i y_i##


    Then (1) becomes ##X - 2t Z + Y t^2 \ge 0##

    and the discriminant, ##D = 4Z^2 - 4XY##


    BUT ... how do we proceed from here .. ... and exactly what properties of the discriminant do we use ...


    Help will be appreciated,

    Peter
     
    Last edited: Aug 7, 2017
  2. jcsd
  3. Aug 7, 2017 #2

    pasmith

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    Homework Helper

    If [itex]ax^2 + bx + c \geq 0[/itex] for all [itex]x \in \mathbb{R}[/itex], then its roots are either both zero or a complex conjugate pair.
     
  4. Aug 8, 2017 #3
    Thanks pasmith ... BUT ... I am not sure how to use that fact ...

    Peter


    EDIT

    On further reflection, then the discriminant is less than or equal to zero ... so the result follows ...

    I think that that is correct ...

    Thanks ...
     
    Last edited: Aug 8, 2017
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