Question

Trivia: What axiom distinguishes the real numbers \(\mathbb{R}\) from the rational numbers \(\mathbb{Q}\) and what does it state?\[\]

Answer 1

It is a good thing to know.

Answer 2

The completeness axiom

Answer 3

And it is,
"Every nonempty set S of real numbers which is bounded above has a supremum"

Answer 4

... from which we can prove many essential things, such as "Every increasing sequence bounded above has a limit"

Answer 5

James, I have a question here....

Answer 6

There is a theorem, which states that if a is lesser than or equal to b added with epsilon (where epsilon is greater than zero), then it implies that a is lesser than or equal to zero

Answer 7

Now, it is one of the first theorems that appear in most books on analysis. Can you tell me what is its significance

Answer 8

I mean why does it have to appear first

Answer 9

I am sorry, I typed it wrong, it should have been
"There is a theorem, which states that if a is lesser than or equal to b added with epsilon (where epsilon is greater than zero), then it implies that a is lesser than or equal to b"

Answer 10

that sounds wrong. For example, if
\[ a \leq 1 + \epsilon \ \ \ \forall \epsilon \]
that only implies
\[ a \leq 1 \]

Answer 11

\[ \forall \epsilon > 0 \]

Answer 12

right!

Answer 13

My question is, "is there anything important about this theorem"

Answer 14

Off the top of my head, I can't think of how it is pivotal in important results. But it is a basic property which I'm sure we use all over the place in epsilon-N or epsilon-delta proofs, most of the time without even thinking too much about it.

Answer 15

And can you suggest a good book for analysis, for beginners ?

Answer 16

Calculus, by Spivak. A book, which the introduction of the later editions admit, would have been better titled "Introduction to Analysis"

Answer 17

Ok thanks a lot.

Answer 18

It's one of my very favorite books in mathematics. It's very well written and does a great job motivating the definitions that we use.