# 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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### How do I prove that $\dfrac{1}{4} \geq r-r^2$ for $r$ a real number between $0<r<1$

I can obviously look at the graph or take the derivative of the function, but I need a more direct and logic based proof. I'm basically just having trouble showing that the value of $(r-r^2)$ ...
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### What is wrong with representing an arbitrary natural and odd square number as $(2n-1)^2$?

Could I not represent every odd square number in $\mathbb N$ using the following notation: $(2n-1)^2$ where $n \in \mathbb N$. For every $n= 1,2,3...$ I get the set $1,9,25...$ every odd square ...
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### Show that $|x|=\frac{\pi}{2} - \frac{4}{\pi}\sum\limits_{k=0}^\infty\frac{\cos\left((2k+1)x\right)}{(2k+1)^2}$

The objective is to find the Fourier series for $|x|$ in the range $-\pi \le x \lt \pi$ so I started by finding the Fourier coefficients: ...
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### Direct proof of the existence of Strong Induction using the Well Ordering Principle

I'm asked to Deduce the alternate form of PMI from WO as a homework problem. To me, this sounds as if I should be doing some form of direct proof of its existence, however, every proof I see that the ...
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### Uniform Convergence of Maximum of Sequence of Functions

Let $K$ be a compact metric space, and $\{f_n\}_{n \in \mathbb{N}}$ is a uniformly bounded, equicontinuous family of functions. Define $$g_n(x) = \max \{f_1(x),f_2(x),\ldots,f_n(x)\}.$$ Prove that ...
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### Relations of natural numbers proof [duplicate]

1) I know that I have to prove three conditions, reflexive, transitive, and antisymmetry but I'm not sure how to start 2) I know that I have to prove there exists a minimum element in each subset, ...
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### Convergence of $\sum_{k=0}^{\infty}{x^k\over (k+1)!}$

How can I prove the convergence of $$\sum_{k=0}^{\infty}{x^k\over (k+1)!}$$ and what is the limit function? I think that I need to use the fact that $\sum_{k=0}^{\infty}{x^k\over k!}$ is a ...
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### Proving $\mathbb{Z}[\sqrt {10}]$ is not a UFD

I am wondering how to show that $\mathbb{Z}[\sqrt {10}]$ is not a UFD. My only idea is to show that there are two factorizations of $10$, say, $ab, uv$ such that $a$ is not a unit times $u$ or ...
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### Proof Help in Real Analysis with sets dealing with Irrational Numbers

The Question: Let $\mathbb{I} \subseteq \mathbb{R}$, which $\mathbb{I}$ is the set of irrational numbers. Prove if $a < b$, then there exits $x \in \mathbb{I}$ such that $a<x<b$. (HINT: ...
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### Proving the divergence of a limit

I need help proving this limit diverges $$\lim_{n\to \infty} n - 2\sqrt{n}= \infty$$ so $\lvert n - 2\sqrt{n}\rvert > M$ so for $n\ge2$, $\lvert n - 2\sqrt{n} \rvert = n - 2\sqrt{n} >M$ ...
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### Problem regarding the connectedness of the Topologist's Sine Curve

$$X=A\cup B=\{(x,\sin({\pi \over x}): 0\lt x\le 1\} \cup \{(0,y) : -1\le y\le 1\}\ \subset \mathbb R^2$$ is called the Topologist's Sine Curve - I . Now what is proved is that $X$ is ...
$$L = \inf\limits_{n\ge1} b_n$$ where $$b_n = \sup\limits _{k\ge n} a_k$$ and $$\{a_k\}_ {k =1}^{\infty}$$ I need to show that if $\epsilon$ is any positive number, then there is an integer N ...