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I am currently reading Spivaks Calculus book 4th edition, and am on the chapter with limits. Theorem one of this chapter says that as $x$ approaches $a$, you can not approach two different limits $l$ and $m$ unless they are equal. The proof of this theorem makes sense except for one thing. He ends up with a contradiction, but I have no idea what he contradicted. Would someone be able to explain it to me? I know I am a little vague, but their are pdf's of his book, so I figured it could be googled.

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In the last line of calculations in the proof, look at the leftmost and rightmost expressions: you get

$$|l-m|<|l-m|$$

and this, of course, is a contradiction: no real number can be less than itself.

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  • $\begingroup$ Oh I see. Thanks for clearing it up for me. $\endgroup$ – user74636 May 16 '13 at 19:11
  • $\begingroup$ Any time, @user74636 . $\endgroup$ – DonAntonio May 16 '13 at 19:14
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He contradicted the existence of two limits $\ell$ and $m$ with $\ell\ne m$. That is, assuming $\ell\ne m$ results in the nonsensical conclusion that a real number is less than itself.

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