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

[Math Amateur]

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:View attachment 7055
In the above text by Sohrab, we read the following:

" ... ... Observe that for any \(\displaystyle t \in \mathbb{R}, F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0\) and look at the discriminant \(\displaystyle Z^2 - XY\) of \(\displaystyle F(t)\). ... ... "Can someone please explain how one proceeds after determining the discriminant and what principles/properties of the discriminant are used?

Help will be appreciated ...

Peter

=========================================================================================Note: my working so far on the exercise is as follows:\(\displaystyle F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0\)

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

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

\(\displaystyle \Longleftrightarrow \sum_{ i = 1 }^n x_i^2 - 2 t \sum_{ i = 1 }^n x_i y_i + t^2 \sum_{ i = 1 }^n y_i^2 \ge 0\) ... ... ... (1)... now let \(\displaystyle X = \sum_{ i = 1 }^n x_i^2\), \(\displaystyle Y = \sum_{ i = 1 }^n y_i^2\) and \(\displaystyle Z = \sum_{ i = 1 }^n x_i y_i\)Then (1) becomes \(\displaystyle X - 2t Z + Y t^2 \ge 0\)

and the discriminant, \(\displaystyle D = 4Z^2 - 4XY\) BUT ... how do we proceed from here .. ... and exactly what properties of the discriminant do we use ...

 

[Opalg]

" ... ... Observe that for any \(\displaystyle t \in \mathbb{R}, F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0\) and look at the discriminant \(\displaystyle Z^2 - XY\) of \(\displaystyle F(t)\). ... ... "Can someone please explain how one proceeds after determining the discriminant and what principles/properties of the discriminant are used?

Help will be appreciated ...

Peter

=========================================================================================Note: my working so far on the exercise is as follows:\(\displaystyle F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0\)

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

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

\(\displaystyle \Longleftrightarrow \sum_{ i = 1 }^n x_i^2 - 2 t \sum_{ i = 1 }^n x_i y_i + t^2 \sum_{ i = 1 }^n y_i^2 \ge 0\) ... ... ... (1)... now let \(\displaystyle X = \sum_{ i = 1 }^n x_i^2\), \(\displaystyle Y = \sum_{ i = 1 }^n y_i^2\) and \(\displaystyle Z = \sum_{ i = 1 }^n x_i y_i\)Then (1) becomes \(\displaystyle X - 2t Z + Y t^2 \ge 0\)

and the discriminant, \(\displaystyle D = 4Z^2 - 4XY\) BUT ... how do we proceed from here .. ... and exactly what properties of the discriminant do we use ...

If the discriminant is positive then the quadratic equation $F(t) = 0$ will have two distinct roots $t = \dfrac{2Z \pm\sqrt D}{2Y}$. If $t$ lies anywhere between these roots then $F(t)$ will be negative. But $F(t)$ is never negative. It follows that the discriminant is not positive, in other words $D \leqslant 0$. Thus $4Z^2 - 4XY \leqslant 0$, so that $Z^2 \leqslant XY.$

 

[Math Amateur]

If the discriminant is positive then the quadratic equation $F(t) = 0$ will have two distinct roots $t = \dfrac{2Z \pm\sqrt D}{2Y}$. If $t$ lies anywhere between these roots then $F(t)$ will be negative. But $F(t)$ is never negative. It follows that the discriminant is not positive, in other words $D \leqslant 0$. Thus $4Z^2 - 4XY \leqslant 0$, so that $Z^2 \leqslant XY.$

Thanks Opalg ... your post was really helpful ...

Peter