Question regarding a proof in Calculus by Spivak In the proof:

I don't really understand how he was able to go from the step starting with "it follows that..." to "Since this is true for all..."
How does the difference between the two being less than epsilon imply that they are the same?
 A: Recall that $\epsilon$ is an arbitrary positive number. In other words, we know that $\inf\{U(f, P')\} - \sup\{L(f, P')\}$ is smaller than every positive number. Also, $\sup\{L(f, P')\} \leq \inf\{U(f, P')\}$. The only way we can satisfy both these properties is if $\inf\{U(f, P')\} - \sup\{L(f, P')\} = 0$.
A: What's critical here is the quantifier "for every $\epsilon > 0$."  
Spivak has shown that the difference is less than $\epsilon$ for every $\epsilon>0$.  That means (for example) that the difference is less than 0.1, and 0.01, and 1.0e-300, and 1.0e-3000, or any other tiny quantity you want to pick.  If the difference was some nonzero number, then you'd could always pick an $\epsilon$ smaller than that number and derive a contradiction.  
A: Imagine this is the real number line
$--------------$
Let's put $U=U(f,P)$ and$L=L(f,P)$ on there
$---L----------U---$
Now we know that the distance between these two is less than $\epsilon$ 
$---L----------U---$
$---|<---(\epsilon< )--->|--$
Given that the smallest upper area achieved by partition $U^*$ is smaller than $U$ and the largest lower area achieved $L^*$ is larger than $L$ we see that
$---L-L^*----U^*-U---$
$---|<---(\epsilon< )--->|--$
Meaning that
$U^*-L^*< \epsilon$
