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.
In the last line of calculations in the proof, look at the leftmost and rightmost expressions: you get
and this, of course, is a contradiction: no real number can be less than itself.
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.