# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### GCD Direct Proof

I want to try and prove this directly because I think it will be more straightforward then a indirect. Also, I believe this has something to do with relatively prime numbers. The help is appreciated! ...
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### I.V.T Continuity proof

If $f$ is defined on $[a,b]$ and has the property that, for any $k$ between $f(a)$ and $f(b)$, there is some $c \in (a,b)$ such that $f(c) = k$, then $f$ must be continuous on $[a,b]$. True or False? ...
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### When there is a proposition $(P\rightarrow Q)$, which row in the truth table of $\rightarrow$ should I use?

I solved one question in a book of analysis, and although I used an informal method to check it, I'd like to know more about what should be done. The question was the following: $A\subset X$ ...
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### Suppose $H:= \{\sigma \in G| \sigma(1) = 1\}$, if for any $j \in \{1,2,…,n\}$ $t_j\in G$ such that $t_j(1) = j$. Show that $|G| = n|H|.$

Let G be a subgroup of the symmetric group $S_n$ in n letters. Consider the following subset of G: $$H:= \{\sigma \in G| \sigma(1) = 1\}$$ Suppose that G acts on the set $\{1,2,...,n\}$ transitively ...
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### Poincare Inequality for 1-Dimensional Problem.

I am referring to the book Introduction to Functional Analysis to Boundary Value Problems and Finite Element by Daya Reddy (page ...
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### Is $T:= \{g \in A_4|g^2 =(1)\}$ a subgroup of $A_4$?

Consider the subset $$T:= \{g \in A_4|g^2 =(1)\}$$ of the alternating group $A_4$ in four letters. Is T a subgroup of $A_4$? My Proof: Yes. If I am not wrong T is the Klein 4-group since only ...
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### Proving the limit comparison test

I have the next attempt: Because $0<L< \infty$, we can find two positive and finite numbers, $m$ and $M$, such that $m<L<M$. Now, because $L = lim_{n\to \infty} \frac{a_{n}}{b_{n}}$ we ...
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### Prove $|A| \le|C|$ for injection and surjective functions

$A$, $B$ and $C$ are finite sets with $F: A \to B$ a surjection and $G: B \to C$ an injection. Prove $|A| \le |C|$ I could prove it using examples, but not sure how to generally.
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### Prove that $\dfrac{\sin{5x}}{\sin{x}}\in\left({-\dfrac54,5}\right)$

Prove that $\dfrac{\sin{5x}}{\sin{x}}\in\left({-\dfrac54,5}\right)$ for any $x\in\mathbb{R}\setminus{k\pi}$ where $k\in\mathbb{Z}$. I wrote $\sin5x$ as $5\cos^4x\sin{x}-10\cos^2 x\sin^3x+\sin^5x$ and ...
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### Trivials are not easy to prove.

If $x,y\in \mathbb{R}$ and $x\neq y$, then show that there are neighborhoods $N_x$ of $x$ and $N_y$ of $y$ shuch that $N_x \cap N_y=\emptyset.$ I know the result is trivial but trivial things are ...
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### Proof: $\sum\limits_{n=1}^\infty \mathbb E(|X_n|)< \infty \Rightarrow \sum\limits_{n=1}^\infty X_n$ converges almost surely

I was reading this as a Lemma, however my book doesn't provide proof of it: Let $X_1,X_2,...$ be a sequence of random variables, then the expression in the title is true. I'm trying to ...
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### Induction Proofs - Mathematics

How do I show by mathematical induction that $2$ divides $n^2 - n$ for all $n$ belonging to the set of Natural Numbers
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### Combinatorial Argument with Natural Numbers

Give a combinatorial argument to show that all natural numbers c(n,k) = c(n,m) where c stands for combination.
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### Prove that a set with n elements in union with an element not in the set has n+1 elements

Suppose $A$ has $n$ elements and suppose $a \notin A$, prove that $A \cup \{a\}$ has $n+1$ elements. I am pretty sure that I am supposed to use something with one to one and onto functions. It all ...
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### Spivak GENERAL limit law proof

Suppose $f(x) \le g(x)$ for all real $x$ Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$ Let limit for $f(x)$ be denoted by $L$ Let limit for $g(x)$ be denoted by $M$. ...
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### Function bijective proving.

Let $\mathbb{C}$ be the set of all complex number. $z\in \mathbb{C}$ Given a function $$f : \mathbb{C} \to \mathbb{C}$$ $$f(z) = (1+2i)z+5i$$ Prove that it is bijective. First, prove ...
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### Functions that satisfy $f(x+y)=f(x)f(y)$ and $f(1)=e$

My real analysis professor mentioned in passing that there exist functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy all of the following conditions for all $a,b \in \mathbb{R}$: $$f(1)=e$$ ...
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### Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$.

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$. Claim 1: $M\cap N$ is a normal subgroup of G: Proof: $1_G \in M$ and $N$ since M ...
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### Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges

Prove that if lim sup$_{n} n^{2}a_{n} = 0 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ converges. Also assume that the sequence is positive. lim sup$_{n} n^{2}a_{n} = 0$ means that for every $\epsilon$, ...
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### Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges

Prove that lim inf$_{n} na_{n} > 1 \Rightarrow \sum_{k = 1}^{\infty} a_{k}$ diverges The solution proof goes like: lim inf$_{n} na_{n} > 1 \Rightarrow$ there exists an $N \in \mathbb{N}$ such ...
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### Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$ Should I prove this by induction? If so, how should I go about it?
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### Prove that $\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+\cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$ for $n\in \mathbb N$

I want to prove that if $n \in \mathbb N$ then $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+ \cdots+ \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}.$$ I think I am stuck on two fronts. First, I don't know ...
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### Can the choice of epsilon be arbitrary in epsilon-delta proofs?

I've been reading Spivak's chapter on limits and something that I don't feel I understand entirely is how the epsilon is decided upon. It makes sense to me in the context of $\,|f(x)-L|<\epsilon$ ...
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### Proof without using the proof of contradiction

By using the proof by contradiction I can determine that the root of a prime number is irrational. But how can I proof this by using the rational roots test to find rational factors of $x^n - p$. How ...
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### Strong induction on a sequence, proving two functions are equal?

Excuse the poor title, but my understanding is still a little fuzzy. Admins feel free to change it Here is the question from the book. suppose that $f_{0}, f_{1}, f_{2}...$ is a sequence defined ...
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### Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
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### To prove given $r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...
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### How do I prove this using proof by contradiction

There is a set a set $S$ of numbers. i.e. $(s_1, s_2, s_3, s_4, s_5, ..., s_n)$. The average of the numbers in the set is $N$. How do I prove that at least one of the numbers in the set is greater ...
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### Poles of analytic functions are isolated

Can the set of poles of an analytic function $f:G\rightarrow \mathbb{C}$ contain a limit point? I know that the answer is no for open $G$, but after thinking more I have become paranoid about ...
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### Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
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### Product identity for $n^n$

I came across the rather nice identity \begin{align} &&\frac{(-n)^{n-1} \Gamma (n+1)}{(1-n)_{n-1}}&&\tag{1}&\\ \\ &=&\prod _{k=1}^{n-1} \frac{(k+1) n^2}{n^2-k ...
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### Divisibility induction proof - question about fractions

I have a question about the example of divisibilty induction proof. Here's the problem [the expression must be divisible by 8]: $5^{n+1} + 2*3^n + 1 = 8*k$ I know that probably I have to proceed ...
Let $B=\{u_1,u_2,...,u_p\}$ where $B$ is an orthonormal basis for a subspace $W$. Let $v$ be any vector in $W$, where $v=a_1u_1+a_2u_2+...+a_pu_p$. prove that $$||v||^2=a_1^2+a_2^2+...+a_p^2$$ So ...