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I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality.

Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result.

The problem is 18 C of page 18 of prologue part of the book. The book can be found in archive.org.

<Moderator's note: Link removed due to copyright violation.>

Here is my proof

https://ibb.co/dc3rUJ

Can any one tell me from where does Spivak get this substitution x=xᵢ/[√(x₁²+x₂²)] and similar one for y ?

Thanks.Sorry no LaTex. If you don't understand my handwriting please ask. Links will expire after 3 days.

Any help is appreciated.

Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result.

The problem is 18 C of page 18 of prologue part of the book. The book can be found in archive.org.

<Moderator's note: Link removed due to copyright violation.>

Here is my proof

https://ibb.co/dc3rUJ

Can any one tell me from where does Spivak get this substitution x=xᵢ/[√(x₁²+x₂²)] and similar one for y ?

Thanks.Sorry no LaTex. If you don't understand my handwriting please ask. Links will expire after 3 days.

Any help is appreciated.

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