Fix limit definition

[xpple]

Any feedback is welcome.

[beastaugh]

I don't think this change is a good idea.

Firstly, the definition is supposed to tell us the conditions under which a given real number is a limit of a given sequence—it's not a definition of when a sequence has a limit. There's then a whole discussion already in this section about the fact that not all infinite sequences of real numbers have limits. So the addition of the existential quantifier doesn't make sense here.

Secondly, the modified definition adds the condition that the absolute value of x - c is greater than zero. But such a value must always be at least zero (it's an absolute value), and it's also fine if it is equal to zero (this will happen if the limit is rational, for example).

Thirdly, I don't think it makes so much sense to have problems in a historical section.

[timbutton]

I totally understand why xpple makes the suggestion, but on balance I agree with Benedict; especially points 1 and 2. So I'm against making this change.

Small additional point. I wouldn't suggest a complete ban on including problems in a historical section. But I reckon any such problems would need to illuminate the history that's being discussed (rather than just be a nice little math problem).

[xpple]

Thanks for the feedback! I agree that in the current context the existential quantifier is a bit out of place. I didn't want to change too much, so I settled with this. I included the problem because Cantor famously said "Der Reihe hat eine bestimmte Grenze" about Cauchy sequences, which is actually false as we know now. A meta-way of seeing this is by observing that the definition of convergence starts with an existential quantifier, whereas the definition of Cauchy-ness doesn't. In the appendix "the Reals as Cauchy Sequences" the limit definition is referenced, but it's stated that it cannot be used because $\epsilon$ ranges over the reals. This is not the fundamental problem, however. The problem is that the definition presupposes the existence of the limit, which I wanted to emphasize by including the existential quantifier. Anyway, thanks for the work on project!