Tagged Questions

109 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
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A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
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How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
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Question regarding Monoalphabetic Phi Test

I've been asked to prove the following system of inequalities; $$1 \ge \phi(T) \ge \frac{n-k}{k(n-1)}$$ Where $\phi(T) = \sum_{i=1}^{k} \frac{n_i (n_i -1)}{n(n-1)}$, $T =$ some text, $n =$ length ...
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Elementary proofs of inequalities

I was just introduced into elementary proofs of inequalities, my text's explanation however feels incomplete. I did further research on the subject, my question is thus: Prove: If $0 < a < b$, ...
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Prove that a set is finite [closed]

Prove that the set of all integers $n$ such that $$36n^2 \geq n^4$$ is ﬁnite. What is the cardinality of this set?
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Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$ I tried induction theorem, when $n = 1$ it is obviously right. But, say $n=k$, It does not make sense since I cannot ...
84 views

Showing $a \le b$ if $a \le b+\varepsilon$, for all $\varepsilon \gt 0$

So I think this is the last problem I have and I'm not thinking I'm doing it properly. Let $a,b$ be real numbers and suppose for all $\varepsilon \gt 0, a \le b+\varepsilon$. Show that $a \le b$. ...
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proving if $0 \le a \le \varepsilon$ for all $\varepsilon \gt 0$ then $a=0$

Suppose $a$ is a real number and we know that $$0 \le a \le \varepsilon$$ for every $\varepsilon \gt 0$. I need to show that $a=0$. The book I am working out of already has shown by contradiction ...
149 views

Choosing the vector that minimizes this sum related to the rearrangement inequality

The rearrangement inequality states that, for two sets of real numbers $x_1\leq\dots{}\leq x_n$ and $y_1\leq\dots{}\leq y_n$, the sum $\sum_{i=1}^n x_{\sigma(i)}y_i$ is minimized for the particular ...
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Prove that $\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\le3$ [duplicate]

Prove that $$\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\le3$$ where $a$ and $b$ and $c$ are positive real numbers. How to do it? Thank you
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Mean Value Theorem: Real Analysis

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
Prove that at least one of the real numbers $a_1 , a_2 , … , a_n$ is greater than or equal to the average of these numbers
Prove that at least one of the real numbers $\,a_1 , a_2 , … , a_n$ is greater than or equal to the average of these numbers. What kind of proof did you use? I think I should use contradiction but I ...