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Chapter 1 Problem 21

I proved it but I don't understand what was the point of proving it. I can rephrase this theorem:

If you have a rectangle with sides lengths $x_0$ and $y_0$ and you want to lengthen or shorten each side in such way that area of the rectangle changes by no more than $\epsilon$, then you can do so by changing $x_0$ side by less than $\text{min}(1, \frac{\epsilon}{2(y_0 + 1)})$ and $y_0$ by less than $\frac{\epsilon}{2(x_0 + 1)}$.

Spivak Answers book says "See chapter 5" which is about limits, so I would guess it's a way for us ot define $\delta(\epsilon)$ to prove that $\forall \epsilon \exists \delta$ something something.

CrabMan
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    Good question! +1 And yes there is no point of this theorem. Most exercises in Spivak focus on technicality rather than the essence (as you can probably guess the essence here is the product rule of limits that the limit of a product is equal to product of limits under certain conditions and it is based on the intuitive fact that if $a$ is an approximation of $A$ and $b$ is an approximation of $B$ then $ab$ is an approximation of $AB$. – Paramanand Singh Aug 07 '17 at 07:15

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Without going into too much detail (which will come in chapter 5), this is proving that the limit of the product is the product of the limits.

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    More specifically, that $\cdot : \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ (or $\cdot : \mathbb{C}\times \mathbb{C}\to \mathbb{C}$) is continuous. – Michael L. Aug 06 '17 at 14:25