Recent content by jem05

Any favorite quote by a mathematician, or a philosopher, or a physicist?

What yould you answer if a professor asks you, Why are the classification theorems of manifolds so important? Why was the classification of surfaces celebrated?

I have the result that any 1 dim topological manifold is either R or S1. And I have the fact that every 1-dim topological manifold is orientable in the sense of orientation on simplices. i want to get that any 1-dim manifold (smooth) is orientable, where orientability is given by the...

personally, the most elegant proof I've seen is the proof of the change of variables formula. It's crazyyy...

What's the most Beautiful proof of a mathematical theorem you've seen?

ok, got it. it actually is not connected

Hello, I know the hyperreals are not connected with respect to the interval opens (open sets are open intervals) In fact *R is totally disconnected. Is *R connected with respect to the S-open topology (open ball around x of radius r is ((x-r, x+r)) any point in this set has its halo in the set...

Hello, Happy holidays everyone, I'm trying to prove that any infinitessimal can be written as a monotone decreasing sequence; that is, one of its representations as a sequence of real numbers is a mon. dec. seq. I'm really stuck, and i don't even know if it's true. Intuitively, it should...

Homework Statement give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else. Homework Equations ---The Attempt at a Solution i got one, and i proved everything, i just want to make sure what i did is correct: f:x n+1 when x is rational...

yes ofcourse, i meant bn strictly positive. ok, thanks a lot

hello, 2 technicality questions: 1) lim sup an /bn = lim sup an / lim sup bn? 2) if lim sup an /bn is finite does that mean that the sequence {an/bn} bounded? thank you.

Homework Statement 1) does there exist an outer measure where every m* nullset is countable? 2) does there exist a lebesgue steiljes outer measure where every m* nullset is countable? Homework Equations The Attempt at a Solution 1) i got an example for it , turned out to be...

ok, thanks yeah sorry for not being clearer, but yeah i assumed i have no identity for my ring R

Homework Statement Let V a subset of the real line be called a vitali set if V contains precisely one point from each coset of the group of rational numbers. Prove: Homework Equations 1) every lebesgue measurable subset of V is a nullset. 2) V is not lebesgue measurable 3) every set of positive...

yeah i agree with this, but sorry i don't see how this is equivalent to what you did. what is our 1 here? after i get that R has identity, then since {0} is a maximal ideal then R/ {0} = R is a field and I am done