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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
Jan
1
revised Why does the Master Theorem work for this example but not the other?
edited tags
Jan
1
revised An algorithmic approach to constructing the real numbers
deleted 1143 characters in body
Jan
1
answered An algorithmic approach to constructing the real numbers
Jan
1
answered Prove the roots of a complex polynomial are imaginary
Jan
1
comment Prove the roots of a complex polynomial are imaginary
This is false - the polynomial is of odd degree with real coefficients and therefore has at least one real root. Unless you want to show that it has two imaginary roots (but may have other roots too)?
Jan
1
comment How can I express mathematically that a set I is the index of another set N?
@Davi It's technically correct, but if you're indexing with the natural numbers $\mathbb N$ anyway, you should just have a sequence $(N_i)_{i\in\mathbb N}$.
Jan
1
comment How can I express mathematically that a set I is the index of another set N?
@Davi The integers in $N$ are the family. The indices are the indices or the index set.
Jan
1
comment How can I express mathematically that a set I is the index of another set N?
@Davi No, there's not really any convention of using a lowercase letter to refer to an element of your sequence. If it was a matrix you might write $a_{ij}$ for an element of $A$, but never with sequences. You can always just call your sequence $n$ to begin with, though.
Dec
31
answered How can I express mathematically that a set I is the index of another set N?
Dec
31
comment V.I. Arnold says Russian students can't solve this problem, but American students can — why?
@fleablood Well, yes, but then by the principle of explosion it would also be 70.
Dec
28
asked Literature on the convergence of $x_{n+1} = f(x_n)$ in general
Dec
27
comment Prove the map has a fixed point
@Niebla In general if we have $\rho(A(x), A(y))<\rho(x,y)$ - note that the inequality is strict - $A$ can only have one fixed point. Let $a, b$ be two fixed points, then $\rho(A(a), B(b))<\rho(a, b)$, which is a contradiction since both sides of this strict inequality are equal.