# Tagged Questions

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Prove that if: $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ than: $\lim_{x\rightarrow\infty}{\frac{f(x)}{x}}=L$ Assuming $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ we can choose $X_{\epsilon}$ s.t. ...
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### $x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
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### Orbit-Stabiliser Theorem Application

Question Let $G$ be the symmetric group $S_n$ acting on the $n$ points $\lbrace 1, 2, 3, . . . , n\rbrace$, let $g\in S_n$ be the n-cycle $(1,2,3,. . . , n)$. By applying the Orbit-Stabiliser Theorem ...
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### Prove that the dual graph of any (planar) graph is connected

I'd like to know if there's a standard proof that the dual graph of any planar graph is connected (or, if there's a counterexample, I'd like to know that too). I've thought of a proof that might work ...
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### Fourier Series of $f(x)=e^x$ on $[0,\pi)$ as a function of period $\pi$

Can you tell me what you get? I've tried computing it, I've got some result but I don't think it's right since I need to use it for something else and it doesn't work at all... What exactly I'm trying ...
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### Prove that if $S$ has a greatest element $b$, then $b = lubS$.

Prove that if $S$ has a greatest element $b$, then $b = lubS$. These are the definitions I used: Def. Given a partially ordered set ($P, \le$), then an element $b$ of a subset $S$ of $P$ is the ...
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### Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
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### Prove that a function defined on points in a plane is zero

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, ...
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### Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
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### The Toad and Frog Game - Proof by Inducation

Toads and Frogs is played on a 1 × n strip of squares. At any time, each square is either empty or occupied by a single toad or frog. Although the game may start at any configuration, it is customary ...
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### The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
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### Prove that $\lim_{n \rightarrow \infty } \frac{n+6}{n^2-6} = 0$.

My attempt: We prove that $\lim\limits_{n \rightarrow \infty } \dfrac{n+6}{n^2-6} =0$. It is sufficient to show that for any $\epsilon \in\textbf{R}^+$, there exists an $K \in \textbf{R}$ such ...
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### Prove that $\displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4}$.

My attempt: We prove that $$\displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4}$$ It is sufficient to show that for an arbitrary real number ...
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### Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

My attempt: Consider the polynomial $(x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0$. By the rational zeros theorem, we can conclude that $\pm 1$ and $\pm 23$ are the only possible rational solutions*. ...
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### Proving that $(\cos \theta )^p\leq (\cos p\, \theta )$ for $0\leq\theta\leq \pi/2$ and $0<p<1$ through an alternative method?

I'm reading the Berkeley Problems in Mathematics book: Prove that $(\cos \theta )^p\leq (\cos p\, \theta )$ for $0\leq\theta\leq \pi/2$ and $0<p<1.$ I could find other ways to prove it, ...
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### Proving the derivative is $0$ at the extremum and all derivatives are $0$.

The pictures below show the proof that Apostol uses in his book. I can't understand why Apostol introduces the function $Q(x)$ and proves the theorem by contradiction using the sign preserving ...
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### Using inequalities and limits

Is it possible to say: $$If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x).$$ My proof(Edit: Proof is wrong due to ...
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### Is my proof correct for: $\sqrt[7]{7!} < \sqrt[8]{8!}$

I have to show that $$\sqrt[7]{7!} < \sqrt[8]{8!}$$ and I did the following steps \begin{align} \sqrt[7]{7!} &< \sqrt[8]{8!} \\ (7!)^{(1/7)} &< (8!)^{(1/8)} \\ (7!)^{(1/7)} - ...
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### Proving the limits of the sum of two functions is equal to the sum of the limits

I am new to proving in math so I want to know if this informal proof of limits is possible: Theorem: If $\lim_{x \to a}f(x)=A$ and $\lim_{x \to a}g(x) = B$, then $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ ...
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### Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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### Proving that morphism of sheaves is iso iff induced morphism on stalks is iso

Is the following proof sound/does anyone have another more elegant (categorical) proof? The direction $\Rightarrow$ is obvious the "family of stalks"-functor is a functor and functors take isos to ...
I need verification for this proof: Q: Suppose $f: (0,1)\rightarrow \mathbb{R}$ is defined by $f(x) = \begin{cases}\frac{1}{n} & \text{if }\text{x is rational with x} = \frac{m}{n}\text{ in ... 1answer 79 views ### Proofs regarding Continuous functions 1 Q: Let$f: \mathbb{R}\rightarrow \mathbb{R}$be a bounded function (that is, there exists some$M\geq 0$so that$|f(x)|\leq M$for all$x\in\mathbb{R}$). Define a new function ... 1answer 95 views ### Alternatinve proof for the principle of the Iterated Suprema The back of the book gave a proof similar to the proof here Proving principle of the Iterated Suprema, but I proved it following way before I checked the back of the book. Could some one verify this ... 1answer 68 views ### Examples of sets which measure cannot be obtained by discretisation I started reading "An introduction to measure theory" by Terence Tao. On page 23 on a pdf reader (pg 7 in the actual document), we are asked to think of an example of a set$E\subset$... 1answer 108 views ### Showing that$a_n \not \to 17$implies a subsequence$a_{n_k}$that is$\epsilon$far from$17$for some$\epsilon > 0$I want to check my proof for this question: Suppose a sequence {$a_n$} does not converge to 17. Prove that there exists some$\epsilon$> 0 and a subsequence {$a_{nk}$} so that$|a_{nk} - 17|$> ... 2answers 201 views ### Verification of proof of the Sequence of Arithmetic Theorem Suppose$\left\{b_{n}\right\}$is a sequence of real numbers which converges to$M$, so that$b_{n} \neq 0$for each$n$, and$M \neq 0$. Prove that the sequence$\{ \frac{1}{b_n} \}$converges to ... 1answer 236 views ### Proof for$-\sup(A) = \inf(-A)$Let$A$and$-A = \{ -x \mid x \in A \}$be two bounded sets. I have to prove that$-\sup(A) = \inf(-A)$, i did it in the following way and wish to know if it is sufficient:$ \exists x\in A$such ... 1answer 206 views ### Dividing Squares Fails to Invoke Contradiction: Two Elementary Divisibility Proofs$x^2 \text{ is even } \iff x \text{ is even } \tag{Thm 3.12, P76}\text{ Let } x, y \in \mathbb{Z}. \text{ Then } x \;\& \; y \text{ are of the same parity } \iff x + y \text{ is even.} \tag{Thm ...
Is that's all? Thank you. :-) A group $H$ is called finitely generated if there is a finite set $A$ such that $H = \left \langle A \right \rangle$ . Prove that every finite group is finitely ...