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I am wondering if the general approach to these proofs involving absolute values and inequalities is to do them case-wise? Is that the typical approach (unless pf course you see some 'trick')? For example, I have:

Prove that if

|x-x_{o}| < ε/2 and Prove that if |y-y_{o}| < ε/2,

then

|(x+y)-(x_{o}+y_{o})| < ε

|(x-y)-(x_{o}-y_{o})| < ε.

It seems like there are way too many cases, but maybe that's just the way it is. For cases, I see:

x=0, y=0

0 ≤ x, y≤0

oh jeesh .... forget listing them. I am now seeing that there are cases when x_{o}is greater than or less than x (though I have a feeling that x_{o}is supposed to be smaller than x and a positive number, but he (Spivak) did not specify...nor did he specify that ε > 0 though I think that it should be. Seems like these are gearing up for ε-δ proofs.

Anyone have any thoughts on any of these points? This is problem 1.20 from Spivak's Calculus.

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# Proofs: Absolute Values and Inequalities

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