The supremum exists because the set $\{\lvert a_1 - a \rvert,\lvert \dots,a_N - a \rvert, \epsilon\}$ has only finitely many elements, so we may just pick the largest element
@0celo7 I'm not trying to get on your nerves, but "why does this supremum exist" is a question I've been asked so often (whether it was actually trivial or not) that you should get used to stating things like "This supremum exists (as a real number) because the set is finite".
@ACuriousMind Ok. Since $b-a>0$, we may apply Lemma 1.2 to get a positive integer $q$ s.t. $q(b-a)>\sqrt{2}$. But if $qa,qb$ differ by more than $\sqrt{2}$, there must be some integer ($p$) multiple of $\sqrt{2}$ with $qa<p\sqrt{2}<qb$. The result follows upon division by $q$.
The fact that $\sqrt{2}$ is not rational is easy to prove.
@0celo7 I'd do a proof by contradicition - assume there are only finitely many rationals $r_1,\dots,r_n$ between $a,b$, then find a rational between $a,b$ that's none of the $r_i$.
@0celo7 "In many respects it is a perfectly ordinary first course in analysis" and "I assume that the reader has had at least one semester of advanced calculus or real analysis at the undergraduate level."
One semester is not a full year, I don't know what exactly "advanced calculus" is.
@0celo7 I'm not sure what you've shown. You've shown that every sequence of integers cannot converge to a non-integer. How does this show bounded sets of integers have an integer least upper bound?
@0celo7 Take a bound for the bounded set of integers. If it's in the set, it's obviously a least upper bound. If it's not in the set, subtract 1. Repeat until you've arrived in the set, that's the least upper bound.
3 pages in, this English book is more confusing than any math
user54412
@0celo7 It probably assumes some familiarity with limit-based calculus and sequences. The sort of thing you'd see in a course that pauses to show you what Dedekind cuts are, but doesn't really expect you to use them.
user54412
That said, I've never taken such a course, and I didn't feel lost with Carothers.
my math curriculum had a very different order from yours
user54412
for comparison, I went to a nearby college to do lin. alg. and multivariable calc in HS, then freshman year I distracted myself with differential equations and statistics and generally applied things, sophomore year was all algebra, and junior year was analysis (and even more algebra)
@NeuroFuzzy I prefer the Cauchy construction over the Dedekind cuts, actually, :P (because it readily generalizes to the notion of completion of a metric space.
Oh, I think philosophy will be fun, but I can take more math by doing it over the summer.
@0B3 That doesn't mean I get credits for it.
user54412
Philosophy and math are two subjects where it really pays off to have a light course load and be free to just dwell on them.
user54412
@0celo7 Even having official courses is no guarantee you'll get credits. I was fortunate to be given the chance to pass out of those classes, but I never got credit toward graduation from them.
@0B3 In practice, "analysis" isn't always "analysis" :p
If I'm not repeating something you already know*. I have a book called "vector analysis" and there is not one proof in there that a pure mathematician would accept completely!
"multivariable analysis" could mean, "take the gradient of this function". Or it could mean, "prove the inverse function theorem". The latter is much harder and belongs to pure math, and the former is really easy and belongs to a second or third calculus course
@ACuriousMind the point is that dedekind cuts require only set theory, while cauchy sequences also topology...the first construction is therefore easier to generalize to nonstandard models
When you go to grad school and meet people who did undergrad in other countries, you'll find that Americans are much more broadly educated than just about everyone else. This necessarily comes at the expense of depth in one's field, for better or for worse.
If a mathematician decides to become a mathematical physicist after completing his phd, what is he missing that other physicists will have in regards to physics?
@DanielSank You have to see this from my perspective. Sure I know some GR and QM and I can do the "math", but so what? I can't program, I haven't done any interesting projects, I haven't interned at national labs. It seems like all of my classmates have done this!
