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

This is from Spivak, Vol. 4 Page 102-103

Given |x-x_0| < 1, |x-x0| < Epsilon/(2(|y_0|+1))

Also given |y-y_0| < Epsilon/(2(|x_0| + 1))

Prove |xy-x_0y_0| < Epsilon

2. Relevant equations

See above

3. The attempt at a solution

The proof proceeds clearly enough. Using |x-x_0| < 1, he shows that |x| < |x_0| + 1.

Then

|xy-x_0y_0| = |x(y-y_0) + y_0(x-x_0)|

< |x(y-y_0)| + |y_0(x-x_0)|

< (1+|x_x0|)*Epsilon/(2(|x0|+1)) + |y_0|*Epsilon/(2(|y_0|+1))

= Epsilon/2 + Epsilon/2 = Epsilon

So.. Q.E.D., but I do not understand the second term...

How is |y_0|*Epsilon/(2(|y_0| + 1)) = Epsilon/2 ??

Any help would be most appreciated. This is for self-study, so I am without a teacher.

Thanks,

Shelly

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Problem understanding a proof in Spivak Vol. 4

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