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I can't find where the fallacy of the following proof lies!
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George Smyridis
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Quoting from Spivak's own answer book:

“How do we know that $\sqrt{a^2-\epsilon}$ and $\sqrt{a^2+\epsilon}$ exist!? In Chapter 7 we prove (Theorem 8) that every positive number has a square root, but the proof of this theorem uses the fact that $f(x)=x^2$ is continuous, which is essentially what we are trying to prove.”

Hans Lundmark
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  • Why is the continuity involved? Continuity states if the value of the function is equal to the limit of the function! So why should we show that if $x^2=a$ then $x= /sqrt{a}$ only if $x^2$ is continuous? – George Smyridis Sep 23 '16 at 07:30
  • I mean, in order to prove that the $f(x)=x^2$ is continuous we have to previously know that $f(a)=a^2$ – George Smyridis Sep 23 '16 at 07:41
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    What he's doing in Chapter 7 is to use the mean value theorem in order to show that (the continuous) function $f(x)=x^2$ takes on every positive value. Before you know that fact, the use of the square root symbol is meaningless. – Hans Lundmark Sep 23 '16 at 10:47