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Yes.

Probability zero things can still occur... What is the probability of a random number chosen in [0,1] being 0.69? It is 0, yet this can happen. This is the difference between "surely" and "almost surely".

Well not quite, the reals have cardinality strictly greater than that of the natural numbers, and hence so do the extended reals. Well, you can think of it as the notion of increasing indefinitely for the purposes of limit computation, but you can also think of the extended reals as an actual...

Yes, because the notion of a limit on the extended reals need not satisfy all of the properties of taking limits on the reals. To talk about limits and convergence, all you need is a set and a topology on it. And your last line further assumes that the extended reals are a field, so we can...

This is true, given the standard way of topologising the extended reals. (In fact it is trivially true for any topology, but one does need a topology / definition of convergence before such a statement makes sense.)

Re: HSC 2015 4U Marathon - Advanced Level If there are any multiple roots at all (real or not), equality holds, so we have no iff. The statement should probably be: For real polys, show that all three roots are distinct and real iff blah > 0.

Re: HSC 2015 4U Marathon - Advanced Level Here is the most elementary proof I know of the fundamental theorem of algebra. (You may assume that every continuous function attains a minimum on any disk of the form D_r=\{z\in\mathbb{C}:|z|\leq r\}. This means that there exists w\in D_r such that...

Re: HSC 2015 4U Marathon - Advanced Level Writing e=(f+(-f))/2, o=(f-(-f))/2 we can write any function as the sum of an odd and even part. p(k)=p(-k) for all k implies that the odd part of p must vanish at 1,2,...,n. As any odd function also vanishes at zero, this odd part must be the zero...

BoS kebab crawl this summer. Shotty not deso.

Agreed, was just talking about city uni options. My favourite used to be Laziko at Strathfield before it closed down. Have had some great ones at random small places in the actual west.

Prefer uni brothers at usyd. Anyway, I was talking about Sydney as a city.

Good ones are pretty rare, Sydney consistently has the best in Aus because of the bigger middle eastern population etc.

Re: HSC 2015 4U Marathon - Advanced Level You should probably provide more justification for this statement.

Sidebar, good kebabs rank #2 in the list of things I miss most about Sydney, narrowly following the people there I care about.

I would wear makeup for a breakfast kebab.

Re: HSC 2015 4U Marathon - Advanced Level Where p is nonzero we have \frac{p'}{p}=(\log |p|)'=(\sum_i \log |x-\alpha_i|)'=\sum_i \frac{1}{x-\alpha_i}. The result extends to all x by continuity.

Re: HSC 2015 4U Marathon - Advanced Level The "taking the limit" step is the real key. We can pass to the limit immediately from the question statement to get: C=C+C^{-1/k} Which trivially has no real (or complex) solutions. (ie, we have the desired contradiction.) This is all just...

Popular => Not intellectual is completely different to Unpopular => Intellectual.

Where did anyone claim unpopular => intellectual? This is obviously not true.

Haha yeah, I tend to experiment a lot when cooking for myself. I have actually been to Malaysia several times (as well as Mamaks), mee goreng is fantastic.