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10h
comment Show that there is a metric space that has a limit point, and each open disk in it is closed. Collecting examples
@columbus8myhw No. The open disk of radius $1$ around $1$ is not closed
15h
comment vercongent sequences
@RickyDemer Any sequence in an empty space is vercongent, convergent, and married to a unicorn
16h
comment Prove that every integer $n\geq 7$ can be expressed as a sum of distinct primes.
The accepted answer to math.stackexchange.com/questions/1382803/… contains a short proof of Richert's result.
16h
comment From the perspective of the multiverse theory, would maths “work the same” in every possible Universe?
The math should be the same, but not the same theories may be applicable to the same phenomena. Even in our home universe we may have that $1$ drop of water plus $1$ drop of water is still just $1$ (bigger) drop of water. This doesn't show that $1+1=1$ instead of $=2$, it shows that for this phenomenon, addition of natural numbers is not a suitable model.
16h
comment How to prove that any natural number $n \geq 34$ can be written as the sum of distinct triangular numbers?
Interestingly, the same question with primes (so using Bertrand's postulate) popped up today as well, I think
18h
comment Prove that every integer $n\geq 7$ can be expressed as a sum of distinct primes.
Richert's theorem does use positive primes only. What you saw in the proof was that sums of the form $\sum_{k=1}^n\epsilon_kp_k$ with $\epsilon_k\in\{-1,1\}$ cover all integers (of correct parity) in a large interval. Adding $\sum_{k=1}^np_k$ produces all even numbers in a suitable interval where all coefficients are in $\{0,2\}$. Division by $2$ produces the claim.
1d
comment How to show the inequality is strict?
Actually, add $\pm$ (or allow complex $c$)
1d
comment how to prove that $ \lim x_n^{y_n}=\lim x_n^{\lim y_n}$?
Note that $x^y=\exp(y\ln x)$ (or how else do you define arbitrary powers?
1d
comment Some questions about Banach Tarski proof
@idkwptc I think I expressed very clearly that there is only a countable number of poles. There is a $2:1$ map from the set of poles to $H\setminus\{1\}$
1d
comment Some questions about Banach Tarski proof
@idkwptc If $x$ is a point in $S^2$ and we havve $hx=x$ then if $x$ can be reached "by applying the proper rotation" $h_1\in H$, say, then this is not unique because $h_2=hh_1$ does the same. Therefore the claim "reached in exactly one way" becomes wrong. We need to classify "the $h$ belonging to $x$" by its word structure, but $h_1$ and $h_2$ may differ in this respect.
1d
comment Prove the result on connected sets in complex analysis.
I think the qeustion is: A topological space with a dense connected subset is connected
1d
comment Proof of Pythagorean theorem without using geometry for a high school student?
@LuckyGuy How is your reply to perterwhy's comment a reply to peterwhy's comment?
1d
comment Proof of Pythagorean theorem without using geometry for a high school student?
Let $F$ be a field of characteristic $0$, let $V$ be a vectors space over $F$ with inner product $\langle,\rangle$. Then for $a,b\in V$ with $\langle a,b\rangle = 0$ we immediately verify $\langle a-b,a-b\rangle=\langle a,a\rangle+\langle b,b\rangle$. -- Seriously, such is hardly enlightening and can perhaps even be considered circular. Personally, I like the geometric proof with the square of side length $a+b$ best.
1d
comment If a triangle has 2 sides of equal length, is it isosceles?
By the way, usually two equal sides is the definition of isosceles (as can be seen from the greek origin isos = eqal + skelos = leg)
2d
comment What is the remainder when 50^51^52 divided by 11?
By little Fermat you should firsct compute $51^{52}\bmod {10}$
2d
comment Can't figure it out, why is this answer is right. By what pattern
Also, I'd have a very nice explanation why 5 would be correct :)
2d
comment Can't figure it out, why is this answer is right. By what pattern
I could explain some distinguishing features, but only ex post, and don't see anything compelling to prefer 1 over 6.
2d
comment Curtains and groups
I just like how you describe that this upsets you :)
2d
comment What to know before solving sequence and series problems?
Intriguingly, "most" sequences containing 87,89,95,107 have 152 where the problem statement expects 157
2d
comment Bijective correspondence between $X$ and $X \cup \{a\}$ for an infinite set $X$
Can you show that infinite implies that you have an injection $\mathbb N\to X$?