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## Homework Statement

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:

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 ...

## Homework Equations

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

## 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

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