Hi, I've been studying from Spivak's Calculus. Normally when I have trouble I can just search, but this time I can't find anything (you can tell how extensive this forum is in that I've been registered here for a while and this is my first post). On page 89, the book gives proof of the following:(adsbygoogle = window.adsbygoogle || []).push({});

If

[tex] |x - x_0| < \text{min} \left (1, \frac{\epsilon}{2(|y_0| + 1)}\right) [/tex] and [tex] |y - y_0| < \frac{\epsilon}{2(|x_0| + 1)}[/tex]

Then

[tex] |xy - x_0y_0| < \epsilon [/tex]

The proof shows this:

1. Since [tex] |x-x_0| < 1 [/tex] we have [tex] |x| - |x_0| \leq |x-x_0| < 1 [/tex] so that [tex] |x| < 1 + |x_0| [/tex]

Thus,

2. [tex] |xy - x_0y_0| = |x(y-y_0) + y_0(x-x_0)| [/tex]

3. [tex]\leq |x| \cdot |y - y_0| + |y_0| \cdot (|x - x_0|) [/tex]

4. [tex]\leq (1 + |x_0|) \cdot \frac{\epsilon}{2(|x_0| + 1)} + |y_0| \cdot \frac{\epsilon}{2(|y_0| + 1)} = \frac{\epsilon}{2} + \frac{\epsilon}{2} [/tex]

Where I have a problem is in how he just assumes that [tex] \text{min} \left (1, \frac{\epsilon}{2(|y_0| + 1)}\right) [/tex] is 1, when I saw the minimum I would have thought you'd do a proof by cases.

Then later, I have a hard time understanding how he went from line 3 to line 4. If [tex] |x| < 1 + |x_0| [/tex] and [tex] |y - y_0| < \frac{\epsilon}{2(|x_0| + 1)}[/tex], then only half of line four makes sense. Otherwise, I'm lost. Will anybody help me out?

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# Difficulty understanding particular proof on limit properties (not epsilon/delta)

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