(Advanced Calculus I) Problem to follow.

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(Advanced Calculus I) Problem to follow.

Mathematics
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Suppose that the real number \(\large\rm a\) has the property that for every natural number \(\large\rm n\), \(\large\rm a\le\dfrac{1}{n}\). Prove that \(\large\rm a\le0\).
I'm thinking I need to use the Archimedean Property in some way here... struggling with it though... hmm
how it can be true ok take n=5 so 1/5 is not smaller than zero right ?

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If you look at all the fractions on the right side of the inequality, the larger and larger than n gets, the smaller and smaller the fraction gets overall. So it's getting really really close to zero. So a, which is smaller than those set of values has to be zero or below since the fractions take up all of the positives. Conceptually I think I understand. I'm just having trouble proving it XD
oh! yes you can use your word to prove it use them into language of math . just make two equations and draw the conclusion .
Do I need to do something with the Completeness Axiom? :o Like if I can show that 0 is the infimum of the set of these (1/n)'s, does that get me on the right track? @zzr0ck3r @ganeshie8 Gotta get the big guns up in hereee XD
This is Archimedes all the way
Suppose \(a>0\), then there exists \(n\) s.t. \(\dfrac{1}{n}
lol @zzr0ck3r yeah in fact this is the property itself :)
oh yes just suppose a>0 is correct and you will find that this is not correct then a<0 will be correct that's all.
\(a\le 0\)
Suppose a>0, then there exists n s.t. 1/n
If you try and use some inf argument, you will end up using the Archimedian property anyway ...I think
Because we were told that \(a<\frac{1}{n}\) for all \(n\).
Oh oh oh great stuff! :) For some reason I thought we were contradicting the fact that a>0. Ok ok ok that makes more sense.
assume a>0, then there exist \(s\in N\) s.t a>1/s , which is a contradiction
yeah, it seems awful circular but it isn't :)
Halmos what is that picture? +_+ What is she so exciting about..? Is that... pie?
i felt so but maybe its away to look at it and understand, but i wonder this is not even calculus this is real analysis don't you feel like ur insulting pure math by saying calculus. @zepdrix yeah it is :P i like pies lol
Real Analysis? 0_o I've been lied to? Gasp...
well idk lol it might be just an introduction, im just saying xD
Ya, the book is titled "Advanced Calculus, second edition". They're probably just taking it nice and easy this first two weeks of class :)
it is all calculus. what you call calculus some people call baby math. I know profs that refer to 600 level analysis classes as calculus
I would not call this real analysis. I would call it advanced calculus.
hmm
My defining line is: when it gets hard :)
even \(\epsilon-delta\) continuity/limit proofs I call advanced calculus
I'm taking Abstract Algebra I at the same time as this class. That class has me very excited. Such interesting stuff :OOOO
good for you. I much prefer abstract algebra.
abstract algebra is my favorite topic :D
same, and algebraic topology is the sh*t
geometric topology existed me :3 idk im in love with pure math and sucks in applied math xD
@Halmos it brought you into existence? 0_o
I have taken general topology and algebraic topology. I start computational topology in the fall. That should either be really cool or super lame.
:)
hmmm typo excited me xD
three people that hate applied math and love pure math excites me:)
awesome <3
I got pretty liberal with the word hate.
Ehhh I still love differential equations :) lol
And anything with complex numbers gets me all giddy XD
Then you will like analysis once you get used to it. Booooo
jk, I like it as well. I T.A. for that class. I really just hate grading the HW. Group theory HW is fun to grade.
I ta analysis that is...
oh neato :O
i like geometry also xD
not difeq, I suck at dif eq. But I once had my class rolling in a difeq class. The teacher was trying to explain something and he said "imagine you had a perfect carpenter" I yelled out JESUS!
wth xD
I have worked through some of Elements(100 pages or so), but that is about all I know in geometry...
ok back to more fun questions for us @zepdrix
\c:
well Euclid, Hilbert, spherical,hyperbolic blah blah all kind of geometry i love <3 and @zepdrix yeah share with us :D
Hilbert had his own geometry also. Damn!!! he did that late in the game...
is it just a bunch of hotels in two space? jk
well in fact he modified Euclid elements and make it more neat and added most of logical thoughts of it, but its known for most its euclidean only :P
known is a big word :)
p.s. @zepdrix showing that the inf is 0 is a great idea and in general that is what you will be doing, but I don't think you can without the Arch property in this case, or something that needs the arch prop in order to prove it. ..
Also, this could be stronger, change your statement to Suppose that the real number \(a\) has the property that for every natural number \(n\), \(a<\frac{1}{n}\).
what math course out there is not hard????

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