Recent content by ConfusedMonkey

Yes, it is sufficient to show that inf(S) = 0 but inf(S) does not belong to S.

If sup(S) belongs to S then the proof is trivial, so assume sup(S) doesn't belong to S.

Why not just say that if there is no sequence convering to sup(S) then there is an epsilon nbd about sup(S) which contains no members of S, and this contradicts the fact that sup(S) is the least upper bound.

I want to learn iOS development. I want to learn both objective-c and swift, but I am having trouble deciding which one is the best one to start with. A lot of answers that I google are not helpful at all. My background: I have some programming experience, but I have not dived deep into any...

As mathwonk likes to say, hard work is much more important than a high IQ. I agree with this. Michael Jordan didn't become who he was just because he was genetically gifted. He busted his butt in the gym every day making sacrifices that others didn't make.

But how can you decide beforehand what will be useful? If, at this very moment, I asked you to compile a list of mathematics that we could develop which will be useful for us, I think that your list would be awfully short compared to we do actually find to be useful in the future. If I'm not...

The only thing that I don't like about Courant is that he doesn't use set-builder notation. Other than that, it's an excellent book.

I remember this problem being assigned in a topology course I took a few years ago and it was a pain in the ass. I also remember the proof being significantly shorter than andrewkirks. Not saying his proof is wrong - I didn't read it - but there is a more elegant way to go about the problem...

My first rigorous calculus book was Courant. I made far slower progress than you have with Spivak. Most people find their first go at rigorous mathematics to be very difficult and slow-going. If you're solving the majority of problems successfully and understanding the material then you are...

Ah, yes. I always forget about that very useful result.

I think you need a homeomorphism. You can "transport the smooth structure" of the circle onto the square by taking smooth charts on the square of the form ##(V, \varphi \circ h^{-1})## where ##h## is a homeomorphism from the square onto the circle (in fact I think a local homeomorphism is...

I can solve it like this: I will keep ##\psi## defined how it is in the original post. Let ##M = \phi^{-1}(0,1)##. I've already proven that ##M## is a submanifold and so the inclusion map ##\iota_M: M \rightarrow \mathbb{R}^4## is smooth. The projection map ##\pi: \mathbb{R}^4 \rightarrow...

Let me preface this post by saying that I only have a very cursory understanding of general relativity. I happen to know that if we assume the cosmological principle, then the hypersurface ##\Sigma_t## of the spacetime manifold ##M##, for any positive ##t##, is either a 3-sphere, a...

Homework Statement Consider the map ##\phi: \mathbb{R}^4 \rightarrow \mathbb{R}^2## defined by ##\phi(x,y,s,t) = (x^2 + y, x^2 + y^2 + s^2 + t^2 + y)##. Show that ##(0,1)## is a regular value of ##\phi##, and that the level set ##\phi^{-1}(0,1)## is diffeomorphic to ##\mathbb{S}^2##. Homework...