I'm trying to understand Spivak's proof of the theorem
A function cannot approach two different limits near $a$. In other words, if $f$ approaches $l$ near $a$, and $f$ approaches $m$ near $a$, then $l=m$.
First thing he rewrites the hypothesis according to the definition:
$0<|x-a|<\delta_1 \rightarrow |f(x)-l|<\varepsilon$
$0<|x-a|<\delta_2 \rightarrow |f(x)-m|<\varepsilon$
He says
We have had to use two numbers $\delta_1$ and $\delta_2$, since there's no guarantee that the $\delta$ which works in one definition will work in the other.
Why does he not use two different numbers $\varepsilon_1$ and $\varepsilon_2$? Such that
$0<|x-a|<\delta_1 \rightarrow |f(x)-l|<\varepsilon_1$
$0<|x-a|<\delta_2 \rightarrow |f(x)-m|<\varepsilon_2$
It's two different limits, why does he use the same $\varepsilon$ for both?
Best Answer
Because he can and wants to.
He uses the following logical rule: if you assume $\forall x,\ P(x)$, you can use $P(y)$ for any $y$. So, if you assume $\forall x,\ P_1(x)$ and $\forall x,\ P_2(x)$, you can assume $P_1(x)$ and $P_2(x)$ for any $x$.
However, if you assume $\exists x,\ Q(x)$, you can name something by a letter, say $y$, that you haven't used before, and assume $Q(y)$. Therefore, if you assume $\exists x,\ Q_1(x)$ and $\exists x,\ Q_2(x)$, all you can do is choose two unused letters, say, $y_1$ and $y_2$, and assume $Q_1(y_1)$ and $Q_2(y_2)$.