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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
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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.
Dec
27
comment How can I prove $x_{n+1} = e^{-x_n}$ is convergent?
So to show $x_n\to a$ (where $a$ is the unique fixed point), we use that $x_n$ has a convergent subsequence, show that the limit of this subsequence must be $a$ with the usual argument, then use the decreasing nature of $|x_n - a|$ to prove that the existence of a convergent subsequence implies the convergence of $(x_n)$ itself?
Dec
27
comment How can I prove $x_{n+1} = e^{-x_n}$ is convergent?
@PhoemueX But $e^{-x}$ is only $1$-lipschitzian on $[0, 1]$.
Dec
27
comment How can I prove $x_{n+1} = e^{-x_n}$ is convergent?
@Wojowu Yes. ${}$
Dec
27
revised How can I prove $x_{n+1} = e^{-x_n}$ is convergent?
added 4 characters in body; edited title
Dec
27
asked How can I prove $x_{n+1} = e^{-x_n}$ is convergent?
Dec
26
answered Let $n ∈ N, n ≥ 1$. Prove that $4^n + 6n - 1$ is divisible by $9$.
Dec
25
answered Why do we do mathematical induction only for positive whole numbers?
Dec
23
revised Describe the rational points on $3x^2 + y^2 = 4$
added 2 characters in body