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8h
comment We have $100$ balls (numbered $1,2,\cdots,100$) and $50$ boxes (numbered $1,2,\cdots,50$).
$\Pr(X_i)=1$ should say $\Pr(X_i=1)$.
9h
revised Proving that a poset can be expressed as a union of subchains.
added 230 characters in body
9h
comment Proving that a poset can be expressed as a union of subchains.
@user162369: That said, you seem to be on the right track. A more rigorous statement of why you can join the chains along the antichain is necessary, though.
9h
comment Proving that a poset can be expressed as a union of subchains.
@user162369: StackExchange isn't designed for this kind of interaction; it's based on an ask-question-get-answer interaction model, not extended conversations. Edits like the kind you're making invalidate existing answers. If you want someone to check your work and keep checking it as you modify it, you could try the chat.
9h
comment Proving that a poset can be expressed as a union of subchains.
@user162369: No, because then those two elements of the antichain would be comparable.
9h
comment What is subtraction?
The distinction you're making doesn't seem significant. You're looking at something two equally-valid ways and asking which one is right.
9h
comment Proving that a poset can be expressed as a union of subchains.
@user162369: Try building two smaller sets, one with all elements higher than the antichain removed and one with all elements lower removed, and try to apply the inductive hypothesis to those sets. The case where one of these subsets is the whole set has to be handled separately.
9h
comment Proving that a poset can be expressed as a union of subchains.
@user162369: Sorry, misremembered the proof. I've looked it up now, so the new hint should work better. I'm afraid your modified proof doesn't work, and bof's poset is a counterexample.
9h
comment Proving that a poset can be expressed as a union of subchains.
@user162369: The proof I know works by induction. Try splitting the set along the antichain into "upper" and "lower" pieces.
10h
comment Proving that a poset can be expressed as a union of subchains.
@bof: I tried to come up with an example like that, but I didn't think of anything. I've replaced the example with yours, as yours is much better.
10h
revised Proving that a poset can be expressed as a union of subchains.
Better example suggested by bof
10h
answered Proving that a poset can be expressed as a union of subchains.
1d
comment Existence of a particular monochromatic sequence from a two colouring of $\mathbb{N}$
The counterexample you've given doesn't work, since we have to choose $x > l_m$ for the proof to make sense (it assumes $2x > 2l_m + 1$). That requirement should have been made explicit when $x$ was introduced. That said, I don't see how we're supposed to get $b+w \le x-k_n$ either. The integers $w$ counts don't need to be between $2k_n+1$ and $x+k_n$.
1d
comment What is the area of a 12cm square?
Did you try Google? In addition to a whole page of useful search results, it actually gives you a little interactive solution thing with a helpful picture and results as you type.
Jul
24
comment Is calculus not rigorous?
@mistermarko: What are you bringing in constructivism for?
Jul
22
comment Is isomorphism not always unique?
"if you're looking for a trivial automorphism group, you're going to be disappointed unless you're into finite, simple groups of order 2" - I'm in luck!
Jul
12
comment Does $\frac{|x_{i+1}-x_n|}{|x_i-x_n|}< 1$ implies convergence?
Is the $i$ in that fraction fixed, or is the relation supposed to hold for all $i$?
Jul
6
comment Limits as $\sin(x)$ approaches infinity
The limit of $\sin^2x$ doesn't exist, but that doesn't mean the limit of $\frac{\sin^2x}{x^2}$ can't.
Jul
3
comment How to read this matrix notation
I was thinking "second diagonal", perhaps the antidiagonal or the diagonal right above or below the main diagonal.
Jul
2
comment Prove that $\sec^2x$ is $C^{\infty} $on $(-\pi /2, \pi /2) $?
@Alexander: This method does that.