Recent content by poochie_d

Oops, you're right, that should be P(A_n-1); I made the correction in the original post. Thanks for pointing it out! I don't think there is anything wrong with applying the power set operation a finite number of times. The above recursion, however, requires as its range a set that contains...

I see. I was trying to figure this out on my own using what little knowledge I have of ZFC set theory and I was getting nowhere =( Thanks for the reference to Kunen!

Given a set S (say, the set of real numbers), define recursively as follows: A_1 = S, \quad A_n = P(S) for n > 1 (here P(S) = power set of S). Is this valid? I am worried because the range of this function is not specified. (In fact, I don't know what the range should be... It certainly...

Suppose there were only finitely many x_n such that x_n < r + \epsilon. Would this in any way contradict the given facts? (Think about the definition of lim sup)

You won't be able to prove this by showing that the sequence is nonincreasing/nondecreasing, since you can find contrary examples satisfying the given conditions. Consider, for instance, the sequence a_n where a_n = n + 1 if n is odd, and a_n = n if n is even. (So the terms of the sequence...

The sequence is not necessarily decreasing: e.g. a_n = n satisfies the given conditions but \frac{a_n}{n} = 1 is nondecreasing. I think it would be be more helpful to show that \frac{a_n}{n} is a Cauchy sequence, since being Cauchy is equivalent to being convergent for real sequences.

Hmm... I don't know what filters are... Better go look it up. Anyway, thanks for the quick reply, micromass! *runs off to wikipedia*

In Theorem 26.7 of Munkres' Topology, it is proved that a product of two compact spaces is compact, and I think the author seems to (rather sneakily) use the choice axiom without mentioning it... Could anyone tell me if this is indeed the case? I don't have a problem with the choice axiom, but...

As wisvuze already mentioned, it's hard to compare the two courses since they contain different materials: MAT224 is linear algebra, while MAT244 is an ODE course. In any case, I would guess that you need to put in some work for MAT224 if you struggled in MAT223, since it's a continuation of...

Thanks for the reply, twofish-quant. Well, I was actually thinking about an M.Sc. (hence the word "master's" in the first sentence). I originally had two main reasons, one of which I now realize is irrelevant: first reason was to leave open the door to the possibility that I might find a...

Hi all, Could anyone recommend any Canadian university offering a master's in physics that is likely to accept students with subpar grades in their upper year courses but who do have some research experience? I guess the top tier schools like University of Toronto or McGill are probably out...

Oh, never mind; I figured it out. It turns out the statement I was trying to prove is not true... e.g. If you have T:\mathbb{R}^2 \to \mathbb{R}, \: T(x,y) = x+y, and S = \{(2,0),(0,2),(1,1)\}, then S is linearly dependent but T(S) = \{2\} is not.

Hi all, Here is the problem: If T: V -> W is a linear transformation and S is a linearly dependent subset of V, then prove that T(S) is linearly dependent. Now, I know that the usual proof goes as follows: Since S is linearly dependent, there are distinct vectors v_1, ..., v_n in S and...

Oh I think I get it now. Thanks micromass!

But aren't tuples other than the ordered pair defined as functions, so that the definition of functions as triples would still be circular?