# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### A **proof** for $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ [duplicate]

I need a proof for the inequality: $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ for all natural numbers $t \geq 2$. For $t=2$ both sides are equal. Can someone find a proof for all $t$? maybe ...
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### Prove integral equality $\int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx$ [closed]

How can I prove the following claim for any given continues function: $$\int_{0}^{\pi} xf(\sin(x))dx = \pi \int_{0}^{\frac{\pi}{2}}f(\sin(x))dx$$ Thanks!
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### Hints on showing that a metric space is complete

Let $C[0,K]$ be the space of all continuous real valued functions on $[0,K]$ for $K>0$ and $L\geq0$, equipped with the metric $d$ defined by $$d(f,g)=\sup_{0\leq k\leq K}e^{-Lk}|f(k)-g(k)|.$$ I ...
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### Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
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### Proving a theorem, what is meant by sufficiency and necessity?

I am looking at the proof of a theorem and the proof begins by saying ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What ...
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### Help determining if a finite subset of $\mathbb R$ is closed and bounded.

If $\{A_n \; : \; n \in \mathbb N\}$ is any collection of subsets of $\mathbb R$, with each set $A_n$ containing finitely many numbers, then the union $\bigcup_{n=1}^{\infty}A_n$ is closed and bounded....
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### The limit of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ as $(x,y) \to (0,0)$

In order to prove that the limit as $(x,y)$ approaches to $(0,0)$ of $f(x,y)= \dfrac {x^2 y}{x^2 + y^2}$ is equal to $0$ is wanted to proof: for ever $\beta\gt0$ exists some $\delta\gt0$such that ...
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### About continuity of scalar fields.

Using the usual definition of limits, with "epsilon and deltas", how can I show that if $x=(x_1,\dots,x_n)$ is a vector in $R^n$, and $f\colon J\to R$,where $R$ is the set of real numbers and $J$ is a ...
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### Determine if the following sentence is a proposition

$2^{101}-1$ is a prime. Besides being a prime and not being a prime, is there any other case the answer could be? If there isnt a third case, Then is a proposition correct? but if there is then is ...
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