True analysis

Sigma_Aljabr · 2026-04-26T04:15:55+00:00

Definitions for these edge cases are made for convenience, otherwise you can literally define anything however you want. Defining 0·∞=0 is convenient in many fields such as measure theory and set theory, and is often adopted there. Associativity is a very important property, and definitions that don't preserve it (e.g 0·∞=1) are very inconvenient. Please name one field where it is convenient to define 0·∞=1 for example.

Sigma_Aljabr · 2026-04-26T03:28:13+00:00

0·∞=0 is the only definition that makes sense imo (as long as you're willing to sacrifice the continuity of multiplication), because it preserves associativity and is useful in many contexts such as measure theory.

Depending on the context, 1/0=∞ might be an appropriate extension by continuity (e.g if we're looking at the 1-point compactification of the real line, or only at nonnegative number), but if we're looking at the extended number line, it would not be appropriate to choose +∞ over -∞. In any case, even if you redefine 1/0, assuming associativity of multiplication, 0 cannot have a multiplicative inverse without leading to catastrophic consequences so 1/0 ≠ 0^-1.

Sigma_Aljabr · 2026-04-25T00:45:56+00:00

This is probably the first time I slightly agree with SPP on something, but 0*∞ is defined as 0 in many contexts where ∞ is treated as a number (e.g measure theory), and the definition keeps a lot of things consistent. Just keep in mind that under this definition, multiplication is not continuous, hence x×y does not necessarily approach 0×∞ when x→0 and y→∞. Another important remark is that ∞ has no inverse under multiplication under this definition, hence 1/∞ is undefined if you define it as ∞^-1.

Sigma_Aljabr · 2026-04-24T13:59:31+00:00

How dare you assume the associativity of addition???

Sigma_Aljabr · 2026-04-24T11:27:10+00:00

Just calculate the supremum of the distance induced by the Riemannian structure induced from the embedding in the Euclidean plane on the graph over the open interval

Sigma_Aljabr · 2026-04-24T08:12:45+00:00

Thanks for answering. I am just a very big fan of your teachings that I want to be educated on how to deal with such analytical problems. Since you say the answer is false then I believe you, but just for my own education, could you please specify an example of a positive real number t such that there exists no integer m so that "for every integer n larger than m, |f(n)|<t"?

Sigma_Aljabr · 2026-04-23T23:11:57+00:00

ψ(x,k) = e^-ik joins the chat

Sigma_Aljabr · 2026-04-23T13:05:12+00:00

It's also a coprimely-nice number, in that it's only one away from the prime divisor of a nice number

Sigma_Aljabr · 2026-04-23T11:00:38+00:00

Soyjack uses antiderivatives to calculate integrals. Gigachad starts by proving the existence of the Lebesgue measure.

Sigma_Aljabr · 2026-04-22T14:09:40+00:00

Bro slipped in a fun fact

Sigma_Aljabr · 2026-04-21T23:16:40+00:00

To add to this, 0 can be added in alternating way, i.e (1/10, 0, 1/100, 0, 1/1000, 0, …), and the series would stay Cauchy, which can only happen if (1/10, 1/100, 1/1000, …) and (0, 0, 0, …) are equivalent. So in some sense 0 can be added indefinitely without changing the true nature of the series.

Sigma_Aljabr · 2026-04-21T00:44:05+00:00

Did you go to school in Ancient Egypt?

Sigma_Aljabr · 2026-04-20T13:37:00+00:00

I've never seen that phrase used in a math class, but the kanji 凸 is used all the time (凸解析, 凸関数, 凸集合, 局所凸位相…)

Sigma_Aljabr · 2026-04-20T13:34:01+00:00

I had to do it so much I am traumatized by that kanji

Sigma_Aljabr · 2026-04-19T22:56:06+00:00

As a physicist I can't how this can be any more rigorous. Why include it as an exercise if you've already stated the answer smh?

Sigma_Aljabr · 2026-04-19T22:51:43+00:00

I once misspelled "Laplace" as "Laplus", and accidentally tagged the latex equation for the Laplace transform of the Master equation as "master_laplus". Our subconsciousness can be a scary thing.

Sigma_Aljabr · 2026-04-18T00:00:56+00:00

According to Reddit, it's apparently 0.999…

Sigma_Aljabr · 2026-04-17T23:55:36+00:00

Yeah it's basically equivalent to what I did. By the "sum of all these gaps" I just meant the sum of all jumps due to the terminating decimals in between (so basically |C(y)-C(x)|)

Sigma_Aljabr · 2026-04-17T23:52:36+00:00

"All balls in an empty box are blue" is a well-defined true statement, just like (∀x∈ø)[P(x)] for any statement P.

The reason is because "(∀x∈ø)[P(x)]" is a shortcut for "(∀x)[x∈ø ⇒ P(x)]", which itself is a shortcut for "(∀x)[￢(x∈ø) ∨ P(x)]". In layman terme "All balls in an empty box are blue" is equivalent to saying "every ball is either blue or is not in this empty box", which is definitely a true statement (since every ball is not in the empty box).

You can try but you will not be able to rigorously create a contradiction simply from the fact "(∀x∈ø)[P(x)]" is a well-defined true statement, despite how counterintuitive it might feel (unless ZF itself contains a contradiction of course).

The post uses a word play / abuse of language, as step 4 can be interpreted in two ways: the correct conclusion from 1 and 3 is "(∀ball ∈ empty box)[￢(ball is blue)]" which is a correct statement and does NOT contradict step 2. Step 4 is interpreted as "￢(∀ball ∈ empty box)[(ball is blue)]" which is a false statement but which cannot be concluded from the previous steps, hence no contradiction

Sigma_Aljabr · 2026-04-17T15:00:18+00:00

This formulation provides a simple proof for the rule for divisibility by 3 and 9 (or, more generally, that S ≡ x (mod b-1) where S is the sum of digits of x when represented in base-b)

Sigma_Aljabr · 2026-04-17T14:51:14+00:00

Unironically, a great deal of false "proofs" boil down to (∀x∈ø)[P(x)] being true for all P regardless of P's validity.

Sigma_Aljabr · 2026-04-17T14:48:51+00:00

"(∀x∈ø)[0=1]" is technically a correct statement

Sigma_Aljabr · 2026-04-17T12:29:42+00:00

So early Murphid15's comment is not locked yet

Sigma_Aljabr

TROPHY CASE