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35s
comment Questions concerning upper densities
Your three questions are incomprehensible to me. Could you kindly restate them in human language, i.e., using words?
3m
revised Questions concerning upper densities
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41m
comment Give an example of an open bounded set whose boundary has no (Lebesgue, Jordan) measure zero
That's a good hint. What did you do with it?
2h
answered How can I prove that without further assumptions Chebyshev's Inequality can not be improved?
3h
comment what is the maximum order one element could have in permutation group $S_5$?
How many of the permutations in $S_5$ have you written down and figured out their order, and what the maximum order you've found so far?
3h
comment Sets of binary sequences
If $(a_n)$ is an aperiodic sequence, then $(1-a_n)$ is also an aperiodic sequence, right? Are those two aperiodic sequences linearly independent?
3h
comment Nowhere dense set and boundary points
"and $X$ not empty" is redundant; if $X$ is empty, then for sure all points of $X$ are boundary points.
3h
comment prove or show that
What does that stuff you wrote have to do with the question? Where did $a_n$ and $b_n$ come from? What happened to $x,y$ and $t$? Did you accidentally copy your answer to some other problem??
3h
comment prove or show that
Way too complicated. You don't need an $\varepsilon$ proof here! If $x\lt y$ and $t\ge0$ then $tx\le ty$, so $x=tx+(1-t)x\le ty+(1-t)x.$
4h
comment How to show that the product of two irrational numbers may be irrational?
@user2357112 Depends. Who's "we"?
6h
comment How to show that the product of two irrational numbers may be irrational?
@HenningMakholm: Oops. Thanks for the correction.
14h
awarded  Nice Answer
16h
comment A table with m rows and n columns
Here's a better version: A table with $m$ rows and $n$ columns is filled with nonnegative integers such that each row contains at least one positive integer. Moreover, if a row and a column intersect in a positive integer, then the sum of the integers in the row is $\ge$ the sum of the integers in the column. Prove that $m\le n$.
18h
revised Conjugate permutation and “their” $\alpha$
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18h
revised Conjugate permutation and “their” $\alpha$
added 3 characters in body
18h
revised Conjugate permutation and “their” $\alpha$
added 9 characters in body
22h
comment If $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function such that $f(x)=0$ for only finitely many values of $x$,
@VHP I thought I explained why (a) is always true in my last comment? You can easily draw an infinite number of nonintersecting continuous paths between those two points (where $f$ has opposite signs), and $f$ has at least one zero on each of those paths, so $f$ has an infinite number of zeros.
23h
comment If $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function such that $f(x)=0$ for only finitely many values of $x$,
The function $f$ is defined on the whole plane, right? There are lots of continuous paths from $a$ to $b$. Let $d$ be the distance between $a$ and $b$. For each $D\ge d$ you can draw a circle (two circles if $D\gt d$) of diameter $D$ which passes through $a$ and $b$, and those circles have no other point in common, so you get a continuum of nonintersecting continuous paths (circle arcs) from $a$ to $b$.
23h
comment If $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function such that $f(x)=0$ for only finitely many values of $x$,
Suppose $f(a)\gt0$ and $f(b)\lt0$. Now suppose you draw a continuous path from $a$ to $b$. Does $f(x)$ have to be $0$ for some $x$ on that path? Now, how many nonintersecting continuous paths can you draw from $a$ to $b$?
1d
answered If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable