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I'm a student of mathematics in Germany.


Mar
18
comment Showing that the sequence $z^n$ is normally but not uniformly convergent
It’s not true. For $n ∈ ℕ$, choose $z = \sqrt[n]{2/3}$, then $|z^n - 0 |= 2/3 > ε$.
Mar
16
comment Question about lim sup
This was not the question, @NasuSama.
Mar
16
comment Group categories with only one object with a defined product
What do you mean by $\mbox {Mor}(\mathcal {G} ; \mathcal {G} ) = \mathcal {G}$? Oh, it seems this is edited and maybe it should be $\mbox {Mor}(G ; G) = \mathcal {G}$ or $… = G$ or something. It still doesn’t make sense to me: What do you want to say here?
Mar
16
comment Abstract algebra (Homomorphism)
Can you give your defintions of a ring and a ring homomorphism? Do you know any characterizations for injectivity? Do you know any examples of rings and maybe subrings?
Mar
16
comment Homework: prove there are two distinct integers $m$,$n$ such that $1/m+1/n$ is an integer.
The second one is preferable. Look at what you have done in your first attempt: You claimed “(1) is true, because (2) is false, because (1) holds as you can see by this example.” There’s a lot of redundancy. But not only that, you actually did go about this by saying: “Suppose (1) is false. Then (2) must be true. But (2) is false, because (1) holds as you can see by this example. And since (2) is false, (1) is true.” Just write: “(1) is true as you can see by this example.”
Mar
15
comment How can I complete this proof?
I haven’t looked through all of this, but let me upvote this for effort nonetheless. Seems legit.
Mar
15
comment The purpose to define open set in metric space
So you’ll end up with a discrete metric. But if you don’t give any context, it is hard to say what the purpose of such a metric is. Why are you even considering discrete metrics?
Mar
15
comment The purpose to define open set in metric space
A metric function is usually $X × X → ℝ_{≥0}$, do you mean for your metric to take values only in $ℤ^+$ (or maybe rather $ℤ_{≥ 0}$, since $d(x,x) = 0$ for all $x ∈ X$)?
Mar
15
comment One-to-one and Onto: True or False
Just think about mapping $: → ·$, i.e. two points onto one.
Mar
15
comment Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
There are so many questions out just bluntly stating the problem to be solved and nothing else, with no or maybe one downvote. I find it peculiar to see three downvotes on this one. To me, this question is fine (except for the part where $g$ is not really well-defined).
Mar
15
comment Maclaurin Series application
Maybe this is about $\tfrac{1}{1-x} = \sum_{k=0}^∞ x^k$ for $x ∈ (0..1)$ (which follows from $\tfrac{1-x^n}{1-x} = \sum_{k=0}^n x^k$). Then you already know the MacLaurin series for $f(x)$ and you can conclude $\tfrac{f^{(k)}(0)}{k!} = 1$ for all $k ∈ ℕ_0$, so you don’t need to differentiate to get the derivatives of $f$ at $0$.
Mar
14
comment Convergence of product over all primes
An interesting question arising is whether the limit is then nonzero.
Mar
14
comment Completion of the unit group of a local field
What is an archimedean prime? Is it an archimedean absolute value, i.e. $|•|_∞$? Interpreting absolute values as primes?
Mar
14
comment How unique is the exponential of sets?
It would be wrong or at the very least very careless to state that any set of cardinality $2^{\aleph_0}$ is an exponential object for $ℕ$ to $ℕ$. It’s like stating that any set with cardinalty $6$ is a cyclic group (which of course can go wrong if you choose the wrong multiplication).
Mar
14
comment How unique is the exponential of sets?
Sorry, that was unclear of me: The essential point is that any exponential object consists of two informations: The object itself and the evaluation map. For instance, $ℕ^ℕ$ is isomorphic to $ℝ$ in the category of sets, yet that doesn’t make $ℝ$ an exponential object. You need to give the evaluation map in both cases and only for $ℕ^ℕ$ there is an obvious choice which is why we don’t state it explicitly. If that is what you meant @MaliceVidrine, I apologize. Although I still think your comment misses the essential point of the issue.
Mar
14
comment How unique is the exponential of sets?
I beg to differ, @MaliceVidrine, see the answer by Hurkyl.
Mar
14
comment A Remark on the definition of meromorphic function
And in complex analysis, discrete is often defined/meant to be closed and discrete. So in this case: Meromorphic functions should have a closed and discrete set of poles.
Mar
13
comment Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
It might not be true that $ℝ ⊂ ℂ$ in a set-theoretic manner, but there is a reason we write it anyway, and this reason for this is explained by one grand master here. Being set-theoretically rigorous creates a lot of weird questions which are rather pointless to deal with. And usually set theory isn’t even done axiomatically at university, but naïvely. I recommend this article.
Mar
13
revised How to prove this algebraically?
added 3 characters in body
Mar
13
answered Prove If $A^{2014}$ is invertible, then $A$ is also invertible