I've just started reading Spivak's Calculus text (4th ed.) and am having some trouble on one of the exercises. The problem asks me to prove that if $|x-x_0|<\frac{\epsilon}{2}$ and $|y-y_0|<\frac{\epsilon}{2}$, then $|x-y-(x_0-y_0)|< \epsilon$. I've proven that it implies that $|x+y-(x_0 + y_0)|<\epsilon$ by adding the two given inequalities and using the addition triangle inequality for absolute value, but I can't find a way to apply the subtraction triangle inequality on this problem. Any help would be appreciated, thanks.
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Are you sure the inequality isn't $|x-y-(x_0-y_0)| < \epsilon$? – cats Jul 27 '13 at 18:36
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Ack, typo. Fixed, thanks. – James Pirlman Jul 27 '13 at 18:37
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2Hint: $|y-y_0| = |y_0-y|$ – cats Jul 27 '13 at 18:38
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1Note that $|y_0-y|\lt \frac{\epsilon}2$ and use the same method – Mark Bennet Jul 27 '13 at 18:39
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Got it, thanks! – James Pirlman Jul 27 '13 at 18:41
