I think (a) may be the undergrad student - still young and idealistic :)
Then (b) could be the PhD student who talked to enough people to be aware that you never know what others consider a ball.
Finally, (c) goes to computer because it's the longest one and it's full of Greek letters. I've known a PhD who believed only in proofs starting with "For arbitrary ε > 0, take δ = ...", but even he wasn't as boring as (c).
I chose (a) also, not because of the restatement, but because it proceeded in such an orderly way, without doubling back. E.g., halfway through (c), there's a sentence starting "We would like to find...", which interrupts the deductive steps to insert an end-goal. It seems unnatural.
Even though I chose (a), and like my argument, I find some of the arguments here for (c) convincing as well.
I could not proceed farther than question 1, because I'm kind of tired of real analysis at this point. Certainly not on Sunday morning. ;-)
There's a line in that proof saying r=min{a,b}; normally I take that to mean r is the minimum of a and b (which makes the proof wrong, since not all metric spaces have obvious orderings on their elements. Spaces like the complex plane or the 2d plane, with an appropriate metric, for instance.
I suppose it could mean r is the point in {a,b} such that the ball B_a(x) or B_b(x) has the smallest radius - but that looks more like a human making a notational mistake, particularly given that both 1b) and 1c) use min(,) in a way that seems correct to me (since they're using min on the values of the metric, not the elements of the metric space.
AFAICT, either the prover made a mistake in logic or a mistake with notation - which I reckon makes him or her human.
Then again, it's been years and years since I thought about this stuff, and I was prone to making mistakes all the time when I did, so everything above might well be wrong. My neurons are getting all fuzzy these days.
"...which makes the proof wrong, since not all metric spaces have obvious orderings on their elements..."
Your reasoning here is wrong. The "a" and "b" come from the range space of the metric, which by definition associates a pair of points in the metric space (unnamed, but call it X) with a nonnegative real number.
In short, "a" and "b" are in R, not in the original metric space X, so it's legal to take the lesser of a and b.
a and b are real numbers, namely the radius of two balls. They forgot to define B_a, but typically
B_a(x) := {y in X : d(x, y) < a}
where (X,d) is the metric space.
I agree with you though, I don't think it's option (a)
It's almost certainly (c), and the rest of the ones that end with "and we're done". The first one is probably not (a) because it has a pronoun ("it's"), which would be a pain to program into a natural language process for this limited purpose.