0
votes
3answers
26 views

How do I prove $x^n < x^m$ when $m > n$ and $x > 1$

Title I made an attempt at it here: $x^n < x^m$ when $m > n$ and $x > 1$, $m$ and $n$ are naturals so divide both sides by $x^n$ so $1 < x^{m-n}$ but here i am stuck. Please help!
1
vote
4answers
41 views

Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

I'm trying to prove this by induction but I'm running into some trouble. The base case is $0$, so, $a^0 = 1$, the inequality holds true Being new to induction, I don't exactly know what to do for ...
0
votes
3answers
25 views

Can the proof for the following 4 cases be simplified to 2 cases?

Let $X$ and $Y$ be finite and disjoint sets. Suppose we are required to prove the following: $|X|\ge 0 \text{ and } |Y|\ge 0 \Rightarrow Q $ where $Q$ is some statement. Therefore, I know I need to ...
0
votes
1answer
24 views

Prove this statement (inequality)

$x,y,z$ $\in\mathbb R$ then $|x-y| \leq|x-z|+|y-z|$ Prove this statement. I thought it was the triangle inequality, but I can't seem to end up with the correct order.
1
vote
1answer
30 views

Help to prove the condition that a right half-open interval is not empty

The right half-open interval is defined as: $[a,b) = \{x \in \mathbb{R}|a \le x \lt b\}$ I need to prove: $[a,b) \ne \emptyset \iff a<b$ My attempt: For $\Rightarrow$: $$\begin{align} ...
0
votes
3answers
54 views

In proof by induction, what does it mean when condition for inductive step is lesser than the propsition itself?

My question is regarding the question posed at the end of the proof. My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false. ...
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 ...
2
votes
1answer
23 views

Proof-Writing $\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$

At the end of this message there are two steps that I do not understand. The proof wants to show in the end that : *$\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$ by definition we have ...
1
vote
0answers
26 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 ...
2
votes
5answers
104 views

Hint in Proving that $n^2\le n!$

The problem is to find the values of n such $n^2\le n!$. I have found this set to be {${n\in Z^+| n\le1 \lor n\ge 4}$} I've used a proof by cases to prove the first part ($\forall n\le 1$) which was ...
-2
votes
1answer
29 views
1
vote
4answers
120 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
82 views

Formal proof of: $x>y$ and $b>0$ implies $bx>by$?

Property: If $x,y,b \in \mathbb{R}$ and $x>y$ and $b>0$, then $bx>by$. What is a formal (low-level) proof of this result? Or is this property taken as axiomatic? The motivation for this ...
0
votes
0answers
38 views

Simple absolute value inequality proof

Prove that if $|x-x_0| < $ $\frac{\epsilon}{2}$ and $|y-y_0| < $ $\frac{\epsilon}{2}$ , then $|(x+y)-(x_0+y_0)|$ $< \epsilon$ and $|(x-y)-(x_0-y_0)|$ $< \epsilon$.
3
votes
2answers
82 views

Is this how you prove by induction for inequalities?

the question is here: http://cpsc.ualr.edu/srini/DM/chapters/examples/ex2.3.2.html My solution is as below:
4
votes
2answers
79 views

Using Results in Sobolev Spaces

I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume $1 \leq p ...
1
vote
2answers
36 views

Reasoning why the implication $t - \epsilon \le x \le t + \epsilon$ for $\epsilon \ge 0 \Rightarrow x = t$ holds using sequences.

In texts I've seen the following reasoning used several times: Suppose $t - \epsilon \le x \le t + \epsilon$ holds for $\epsilon \ge 0$. Then it in particular holds for $t - \frac 1 n \le x \le t + ...
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 ...
5
votes
3answers
128 views

How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
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
72 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 ...
2
votes
1answer
55 views

Proving the Schwartz inequality, need some help

The question and my attempt can be found here: http://i.imgur.com/CiMNr2m.jpg?1 I don't quite understand what i'm suppose to do. It says to prove the inequality with whatever so I tried to ...
5
votes
2answers
202 views

Limit superior inequalities proof

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
1
vote
3answers
256 views

Proof for limit superior's property: $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ [duplicate]

Let $a_n,b_n>0$ for all $n\in\mathbb N$. Prove that $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ I know that $\limsup (a_n+b_n ) \leq \limsup a_n + \limsup b_n$. But I don't know how ...
-3
votes
3answers
89 views

Prove that this inequality $5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$

Let $a,b,c>0$ and $a+b+c=1$ Prove that $$5(a^2+b^2+c^2) \leq 6(a^3+b^3+c^3)+1$$
6
votes
4answers
321 views

How to show that a limit cannot be another number?

Let: $$ G(x) = \left\{ \begin{array} {cc} x \sin \frac{1}{x} , & x\neq 0 \\ 0, & x=0 \end{array} \right. $$ I can understand that the function is continuous at $x=0$ because: For ...
1
vote
1answer
45 views

How to find the $l$ in $-\epsilon + 1 < l < \epsilon + 1$?

Suppose we have: $$ \epsilon > 0 $$ and $$ -\epsilon + 1 < l < \epsilon +1 $$ I think I have enough information to show that $l=1$ but I don't know how to formally show that this is true. ...
2
votes
3answers
84 views

Problem with Proof of Inequality with Squares by Induction

I am a bit new to logical induction, so I apologize if this question is a bit basic. I tried proving this by induction: $$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$ Starting with the base ...
1
vote
1answer
104 views

How do I check the triangle inequality using subsets?

How do I establish the triangle inequality $|x+y| \geq |x|+|y|$ by considering all real numbers as the union of six subsets and checking the inequality on each of those subsets?
3
votes
1answer
125 views

Prove $\frac{1}{bc + cd + da -1} + \frac{1}{ab + cd + da -1} + \frac{1}{ab + bc + da -1} + \frac{1}{ab + bc + cd -1} \le 2$

If $a,b,c,d > 0$ and $abcd = 1$, prove that the following inequality holds: $$\frac{1}{bc + cd + da -1} + \frac{1}{ab + cd + da -1} + \frac{1}{ab + bc + da -1} + \frac{1}{ab + bc + cd -1} \le ...
15
votes
4answers
591 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
2
votes
3answers
152 views

Showing that $3n<n!$ whenever $n$ is an integer with $n \geq 7$

How can we show that: $$3n< n!$$ whenever $n$ is an integer such that $n \geq 7$ ? I was thinking that we can prove this by showing that such case is true with any integer above 7, but ...
1
vote
2answers
72 views

Prove $|A| + |B| \ge |A \cup B|$ if $A$ and $B$ are finite sets

I have a question like this: Prove $|A| + |B| \ge |A \cup B|$ if $A$ and $B$ are finite sets. Here is the solution but I don't agree with it: Let $A = \{a_1, \dots, a_n\}$ and $B = \{b_1, \dots, ...
4
votes
4answers
176 views

Working with proofs help?

I'm trying to study for my midterm and doing some random practise questions to work with proofs. However I'm stuck on, as the only way I know how to prove it is through plugging in numbers, however as ...
1
vote
3answers
79 views

Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$

Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$ Here is what I would do if this were asked on a test and I was told to "justify" the answer. Let $x \in ...
4
votes
4answers
174 views

Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.

I am trying to prove this statement for all $ n \geq 1 $ using induction: $$ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3. $$ I said: Base case $ n = 1 $: $$ \left( ...
4
votes
4answers
271 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
10
votes
5answers
499 views

Prove $|a+b|+|a-b| \geq |a|+|b|$

I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$ I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$ How can I use this ...
1
vote
2answers
94 views

Confused where and why inequality sign changes when proving probability inequality

"Let A and B be two events in a sample space such that 0 < P(A) < 1. Let A' denote the complement of A. Show that is P(B|A) > P(B), then P(B|A') < P(B)." This was my proof: $$ P(B| A) > ...
2
votes
1answer
118 views

Is this induction procedure correct? ($2^n<n!$)

I am rather new to mathematical induction. Specially inequalities, as seen here How to use mathematical induction with inequalities?. Thanks to that question, I've been able to solve some of the form ...
1
vote
2answers
92 views

Theorem about two real numbers 2

My question is: Prove- If $a,b$ are two positive real numbers such that their sum is $a+b=k$. Then the product $ab$ is maximum if and only if $a=b=\displaystyle\frac{k}{2}$. I proved the ...
4
votes
1answer
14k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
4
votes
3answers
486 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
1
vote
1answer
990 views

Proof the inequality $n! \geq 2^n$ by induction

I'm having difficulity solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this with induction. I started like this: The lowest natural number where the ...
4
votes
5answers
381 views

Showing $a^2 < b^2$, if $0 < a < b$

Lately, I've been stumbling with proofs of inequalities. For example: Given $0 < a < b$ Show $a^2 < b^2$ The only thing I've been able to come up with so far: $a^2 < b^2$ ...
4
votes
4answers
200 views

Proving an inequality

Suppose $a$ and $b$ are real numbers. Prove that if $a<b$ then $\frac{a+b}{2}<b$. The 'solution' hints at adding $b$ to both sides of the inequality $a<b$, and $a+b<2b$ is as far as I've ...