Jack M
Reputation
12,122
Next privilege 15,000 Rep.
Protect questions
 Jan 4 asked Is something wrong with my confidence interval for a Binomial variable? Jan 4 comment Is there a meaningful distinction between “direct” and “iterative” methods for solving equations? The distinction is exactly what you say in your question: an iterative method is not a formula expressed in radicals. If I say $x=1+\sqrt 2$ I have expressed $x$ using a formula expressed in radicals. If I say "$x$ is the limit of $x_{n+1}=f(x_n)$ when $x_0\in[-1, 2]$, I haven't. Jan 4 comment Does a game need below-average players @FlorianPeschka If we have $n$ players, and they all have greater than $0.5$ win rate, then the sum of their win rates (which must still be equal to $1$) is at least $n0.5$, which is even worse than what we get with $2$ players! Jan 4 comment Is collapsing considered a legitimate proof? @DanielR.Collins The $S$ in this answer has nothing to do with the $S$ in the question. Jan 3 awarded Nice Answer Jan 3 comment Is collapsing considered a legitimate proof? @S.Mo I rewrote the proof to use a structure closer to what you're used to. Jan 3 revised Is collapsing considered a legitimate proof? added 191 characters in body Jan 3 comment Is collapsing considered a legitimate proof? @S.Mo That is a correct proof by induction, but different from the one I gave. The proof by induction I give in my answer is directly modeled on the "collapsing" method you used in your question. Jan 3 comment Is collapsing considered a legitimate proof? @S.Mo In induction we have a predicate $P(n)$, that is, a true-false statement into which we can plug $n$ like a variable, for example, an equation involving $n$. We then show that if $P(n)$ is true for any $n$, then $P(n+1)$ is true. Using phrasing like "show that it resembles" makes it seem like induction is just a method of shuffling symbols around. Jan 3 answered Is collapsing considered a legitimate proof? Jan 3 comment Prove the roots of a complex polynomial are imaginary @inya Do you understand basic polynomial algebra concepts like polynomial long division, the remainder theorem, etc? Khan Academy's "Polynomial Arithmetic" course covers it, or you could look for a textbook on (modern) algebra and skip to the chapter on polynomials. If you understand all these concepts then I think the Wiki article on the Polynomial GCD should be sufficient if you soldier through it. Jan 3 comment My proof that $S_n/\sqrt n$ does not converge in probability I don't understand your construction though. To what are you claiming your $S_n/\sqrt n$ converges in probability? Jan 3 comment My proof that $S_n/\sqrt n$ does not converge in probability In response to your last paragraph: Yes, I think for this reason this proof (were it valid) would necessarily have to be posed as a proof by contradiction (unlike many proofs by contradiction which can be rephrased as direct proofs of contrapositives). You have to assume the sequence converges in probability to $Z$ in order to even be able to say what $Z$ is. Jan 2 accepted An algorithmic approach to constructing the real numbers Jan 2 comment My proof that $S_n/\sqrt n$ does not converge in probability @Winther I don't think that really answers the question - my question is specifically about whether the approach I used is valid or can be modified to be valid. Jan 2 comment Does a connected countable metric space exist? Very intuitive proof. You're using a result that says the only connected subsets of $\mathbb R$ are the intervals, I guess? Jan 2 comment An algorithmic approach to constructing the real numbers So you're saying there's no way to patch up the hole in the algorithm I proposed in my answer? Jan 2 asked My proof that $S_n/\sqrt n$ does not converge in probability Jan 2 comment What is the fundamental matrix solution? Do you know about matrix exponentials? Jan 1 awarded Nice Answer