(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am using Spivak's Calculus and just finished the third exercise in part 1.

It was a very easy exercise but it seems that Spivak makes some assumptions.

The problem is as stated:

If x[itex]^{2}[/itex]=y[itex]^{2}[/itex], then either x=y or x=-y. Prove it.

The proof was relatively simple (by factoring out (x-y)(x+y) from x[itex]^{2}[/itex]

-y[itex]^{2}[/itex] and showing that either (x+y)=0 or (x-y)=0 or both). What I had problems with was that it was assumed that a(0)=0. So, I will now propose a proof that a(0)=0 for all real a using the properties listed in the book. All I need is a confirmation that my proof haw no flaw. Thank you.

2. Relevant equations

3. The attempt at a solution

Introduce (a*0).

(a*0)+(a*0)=0+(a*0).

By the distributive law

a(0+0)=(a*0)+0

By the existence of the additive identity

a*0=(a*0)+0.

Subtracting (a*0) from both sides yields

(a*0)-(a*0)=0

And by the existence of additive identity

0=0

Which is true. Is this valid?

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# Verification of Simple Proof

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