4
votes
2answers
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: ...
0
votes
0answers
38 views

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 ...
0
votes
1answer
41 views

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 ...
4
votes
1answer
75 views

how to solve this elementary induction proof: $\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\le\ 2-\frac{1}{n}$

this is a seemingly simple induction question that has me confused about perhaps my understanding of how to apply induction the question; $$\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\ \le\ 2-\frac{1}{n},\ ...
3
votes
3answers
49 views

Prove that $2n+1 \leq 2^n$ for $n \geq 3$ using mathematical induction.

Question: $2n+1 \leq 2^n$, for all $n \geq 3$ I've tried: Basis: $P(3) = 7 \leq 8 $, so basis step is valid Pick an arbitrary value from the universe, $k \geq 3$ Inductive Step: $2k + 1 \leq ...
0
votes
1answer
46 views

Proof of AM-GM Inequality with lemmas

I need to prove the AM-GM Inequality using a few specific lemmas that I have already proven. I'm mostly just unsure what to do next and how to tie it all together at the end to finish the proof. Here ...
3
votes
4answers
80 views

How to easily prove $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$ [duplicate]

When I tried to solve some certain math problem (an inequation) for pivate exercise purposes, I had to prove that $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$, I solved it with tools from differential ...
0
votes
2answers
62 views

How would this problem need the Mean Value Theorem?

I'm asked to square the inequality and use the Mean Value Theorem to prove that $$\sqrt{1+x} < 1 + \frac{x}{2}$$ for $x>0$. Unfortunately, I don't really understand why I would need such a ...
0
votes
0answers
19 views

complete logic for proving inequalities

Last semester I took a course on algorithm analysis a big part of which was proving that the running time function of a program was in the set $O(f(x))$ for some $f$. To prove $f\in O(g(x))$ one ...
1
vote
1answer
24 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
1
vote
0answers
30 views

Can proof by contradiction and counterexample by used at the same proof?

Here is a part of a theorem: If $\alpha>1$ and $x\ge-1$ then $(1+x)^\alpha \ge 1 + \alpha x$ I was wondering if I could use proof by contradiction and counterexample at the same time. Assume ...
1
vote
1answer
87 views

Cauchy-Schwarz Inequality - Proof using Quadratic Polynomial [Lay P379 Thm 6.7.16]

I don't perceive https://www.dpmms.cam.ac.uk/~wtg10/csineq.html, about why it " is an obvious thing to write down" "a quadratic form, use the fact that it is non-negative everywhere, and ...
1
vote
2answers
29 views

show that there is not positive integer n satisfying $4n<n^2<5n$

I do not know where to begin, I know that if I choose some n, to be say n=3, this does not hold, and it doesn't hold for any n > 0. But that's not a formal mathematical proof.
2
votes
1answer
91 views

Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
0
votes
1answer
23 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
-2
votes
1answer
29 views
0
votes
1answer
31 views

Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$ \frac{a}{b} < \frac{c}{d} $$. Prove that: $$ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $$. I would like to find an intuitive way to solve ...
0
votes
2answers
41 views

Solving Problem by different Method ( non-induction)

I have this problem , which I was able to prove it by induction, but I wonder could be solve by direct method ( for example combinatorial method). I want to find number of solution for $$0 \le ...
1
vote
1answer
23 views

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 ...
0
votes
4answers
83 views

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 $0 < a < b$, ...
-1
votes
2answers
43 views

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?
0
votes
1answer
60 views

Very complicated limit and trying to find convergence

I have no idea how to prove this: $\lim_{n \to \infty} \frac{2^2 \times4^2\times6^2\times\dots\times(2n)^2}{(1\times3)(3\times5)\dots((2n-1)(2n+1))} = \lim_{n \to \infty}\frac{2^2 ...
0
votes
1answer
48 views

Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$

Prove the following: $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$ Hello everyone, I am trying to create some sort of ...
2
votes
6answers
98 views

Prove inequality and when does equality hold?

Prove that for any real positive numbers $a$ and $b$ $$\left(a+\frac{1}{b}\right)\left(b+\frac{1}{a}\right) \geq 4.$$ When does equality hold?
2
votes
1answer
166 views

Proving that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$ for all positive $n$

Hello I am having difficulty proving the following inequality: $$ \sqrt[n]{n} < 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n. $$ I am trying to use mathematical induction ...
0
votes
2answers
58 views

Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$

Hello I have been blasting at this inequality proof and it is just not doing what I want it to do: Prove that $2^{n-1}(a_1a_2\ldots a_n + 1) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ assuming that ...
2
votes
1answer
36 views

Inductive proof of an inequality

I am trying to prove this inequality by induction: For all $x$ in the interval $x\in [0, \pi]$, prove that: $$ |\sin (nx)| \leq n\sin(x) \textit{, n a nonnegative integer}$$ The base case is ...
1
vote
4answers
124 views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq ...
3
votes
3answers
136 views

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 ...
1
vote
4answers
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 $a \le b$. ...
2
votes
1answer
41 views

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 ...
1
vote
0answers
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 ...
1
vote
0answers
70 views

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
2
votes
2answers
191 views

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 ...
3
votes
2answers
86 views

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 ...
3
votes
2answers
524 views

Proving Holder's inequality using Jensen's inequality

Let $p$ and $q$ be positive reals such that $\frac{1}{p}+\frac{1}{q} = 1$, so that $p,q$ in $(1,\infty)$. For $\vec a$ and $\vec b \in \mathbb{R}^2$ prove that $|\vec a \cdot \vec b | \leq ||\vec ...
-1
votes
4answers
62 views

Proving $ \frac{1}{t} < \ln (t) - \ln (t-1) < \frac {1}{t-1}, $ for $t>1$ [closed]

Prove that if $t>1$ then, $$ \frac{1}{t} < \ln (t) - \ln (t-1) < \frac {1}{t-1}. $$ Any kind of input on this problem would be most appreciated!
1
vote
1answer
50 views

Proving inequalities by induction

I'm having trouble understand the inductive when proving inequalities; Here's an example: Show that $2^n \gt n^2 $ for any integer $n \gt 4 $. Well for the basis $n=5$, it shows: $32>25$ Now, ...
1
vote
2answers
45 views

prove that for $n \ge 4, {{2n}\choose{n}} \ge n\cdot2^n$

Prove that for $n \ge 4$ $${{2n}\choose{n}} \ge n\times2^n$$ I tried like that: $T_4$: ${{8}\choose{4}} = 70 \ge 4\times2^4$ = 64 so it's ok $T_{n+1}$: $$\frac{(2n+2)!}{(n+1)!)(n+1)!} \ge ...
5
votes
2answers
295 views

Proving Inequality with the Greatest Integer Function

Show that $$[(m+n)x]+[(m+n)y] \ge [mx+(n-1)y]+[my+(n-1)x]$$ where $m,~n \in \Bbb{N}$ and $0\le x,~y < 1$. I've tried everything for about half a day and still couldn't figure it out. ...
7
votes
3answers
171 views

If $x_1, \ldots, x_6$ are positive real numbers that add up to $2$. Show that:

If $x_1,x_2,x_3,x_4,x_5$ and $x_6$ are positive real numbers that add up to $2$, then: $$2^{12} \leq \left(1+\dfrac{1}{x_1}\right) ...
2
votes
2answers
98 views

How to prove that if $-1<x<0$ then $x^2 + x < 0$?

I am trying to prove an equivalence. I have already proved that: $$x^2 + x < 0 \implies -1 < x < 0 $$ using a sub-proof by cases, in which I used the fact that when $xy < 0$, $x$ and ...
0
votes
2answers
119 views

Proving that $1/x$ and $1/x^2$ limit does not exist

1) If I am to prove that limit of $ \frac1x$ doesn't exist at $x\to0$ is it sufficient and rigorous enough to show that the left hand and the right hand limits are not equal(EDIT: are not equal ...
3
votes
1answer
115 views

Arithmetic and Geometric Mean Inequality

Use the AM - GM inequality (no other method is acceptable), to prove that for all positive integers $n$: $$\left(1 +\dfrac{1}{n}\right)^n \leq \left(1 + \dfrac{1}{n+1}\right)^{n+1}$$ I see that it ...
-1
votes
2answers
98 views

How prove this $\sqrt{x^4+7x^3+x^2+7x}+3\sqrt{3x}+x^2-10x\ge 0$

let $x\ge 0$,show that $$\sqrt{x^4+7x^3+x^2+7x}+3\sqrt{3x}+x^2-10x\ge 0$$ My try: let $a=\sqrt{x^4+7x^3+x^2+7x},b=3\sqrt{3x}+x^2-10x$ so $$\Longleftrightarrow a+b\ge 0$$ if $b\ge 0$ then $a+b\ge 0$ ...
1
vote
2answers
77 views

Let $a,b,c \in \mathbb{R^+}$, does this inequality holds $\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$?

Does the following statement/inequality holds for $a,b,c \in \mathbb{R^+}$? $$\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$$ I've been thinking for hours and I ...
2
votes
0answers
74 views

Prove $\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 + b^2}\right) \ge 9$

If $a,b,c \in \mathbb{R^+}$,then prove that the following inequality holds: $$\left(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\right)\left(\frac{ab}{c^2+b^2} + \frac{bc}{a^2 + c^2} + \frac{ca}{a^2 ...
6
votes
2answers
129 views

Polynomial inequality proof

Prove $(1-a)(1-b)(1-c)(1-d)>1-a-b-c-d$ and $ a,b,c,d$ are real and between 0 and 1. I can do this with $$(1-a)(1-b)>1-a-b \\ 1-a-b+ab>1-a-b \\ ab>0 $$ But with $c$ and $d$, this ...
2
votes
2answers
47 views

Let $a,b$ nonnegative, $a\ge cb$ for every $c$ in $(0,1)$, Should $a \ge b$ be true?

In the proof of monotone convergent theorem, The above technique is used. I really appreciate if anybody can explain it to me.
1
vote
0answers
55 views

Prove that the length of this curve decreases as one of its parameters increases

The following is the problem statement of one of my assignment questions. Consider the $\partial_t (t,s) = K(t, s)N(t, s)$ for all $t \geq 0$, and for all $s \in [0, 1]$, where $T(t, s) = ...