Inequality Definition and 1000 Threads

Homework Statement Show that if n belongs to N, and: An: = (1 + 1/n)^n then An < An+1 for all natural n. (Hint, look at the ratios An+1/An, and use Bernoulli's inequality) The Attempt at a Solution I think i have a vague idea of what to do here, like I am sure induction is involved in this...

Homework Statement Prove that 1 + 1/2 + 1/4 + 1/7 + 1/11 + ...... <= 2*pi Homework Equations none The Attempt at a Solution all i could figure out was the nth term of the sequence T(n) = \frac{2}{2 + n(n-1)} any help appreciated.:biggrin:

Homework Statement solve the following equations or inequalties for x in the interval [0,2pi) 2cos^2(x) + 1 = 3cos(2x) Homework Equations The Attempt at a Solution My attempt at the problem: 2cosx(cos2x) + 1 = 3cos(2x) 2cosx(cos^2x - sin^2x) + 1 = 3cos(2x) 2cos^3x -...

How can we graph this inequality - |y|+1/2>=e-|x| ? I drew the function(actually a combination of functions) for equality. It would be symmetric in all quadrants and intersect the axes at +ln2 and -ln2(x-axis) and 1/2 and -1/2(y-axis).However since the various graphs are mixed up it is hard...

Inequality Solution [URGENT, +Part solution included] Homework Statement l 3/(x-1) - 5 l < 4 Homework Equations The Attempt at a Solution so here's where I am abit confused. since the inequality sign is not > or >= but instead in this case it is <. Therefore, x has to be...

Homework Statement Let x and y be real numbers. Prove that if x =< y + k for every positive real number k, then x =< y The Attempt at a Solution x =< y + k -y + x =< k since k is positive, the lowest value it can take doesn't include 0: -y + x < 0 x < y So I get x < y from x =< y...

Homework Statement l [3/(x-1)] - 5l < 4Homework Equations The Attempt at a Solution My 1st step was to make the inequality like this. -4 < 3/(x-1) - 5 < 4 and then i multiplied (x-1) to both left and right side and as well as to the 5. but in the end, my result turns out to be...

Homework Statement Given 0 <= a <= b show that, a <= sqrt(ab) <= (a+ b / 2) <= b Homework Equations a * b <= a^2 / a*b <= a* a The Attempt at a Solution I think i know where I am going but i wanted to make sure if its correct so far. So we know that...

Homework Statement 1] l x + y l < or equal to l x l + l y l Homework Equations x^2 + 2xy + y^2 The Attempt at a Solution Left side. i Squared left side to begin with, and i got x^2 + 2xy + y^2 and also did the same for the right side, but it would have absolute sign...

Homework Statement From Spivak's Calculus, Chapter 2 Problem 22 Part A: Here, A_{n} and G_{n} stand for the arithmetic and geometric means respectively and a_{i}\geq 0 for i=1,\cdots,n. Suppose that a_{1} < A_{n}. Then some a_{i} satisfies a_{i} > A_{n}; for convenience, say a_{2} >...

I know that we can square an inequality if both sides are positive. But can we cube an inequality provided both the sides are positive? If no then why?

Homework Statement *Sorry I could not get the math symbols to work properly so I did it by hand. I hope this isn't too much trouble. Prove: | sqrt( x ) - sqrt( y ) | <= | sqrt ( x - y ) | for x, y >= 0 Hint: Treat the cases x >= y and x <= y separately. I am new to proofs and we can't use...

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|. How do we know that? is |z+w|=|z-w|?? Note that z and w are complex numbers.

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|. How do we know that? is |z+w|=|z-w|?? Note that z and w are complex numbers.

given z, w\inC, and |z|=([conjugate of z]z)1/2 , prove ||z|-|w|| \leq |z-w| \leq |z|+|w| I squared all three terms and ended up with : -2|z||w| \leq |-2zw| \leq 2|z||w| I know this leaves the right 2 equal to each other but i figured if i show that since there exists a z\geqw\geq0, then...

My problem states: Given z, w\inC, prove: ||z|-|w||\leq|z-w|\leq|z|+|w|. Now, I am confused because, isn't it true that ||z|-|w||=|z-w| ? I am using Rudin's book which gives |z|=([z's conjugate]z)1/2

Homework Statement -3<(1/X)≤ 1 Solve. Homework Equations The Attempt at a Solution Here's my attempt at it: (1/X)≤1 and (1/X)> -3 X≥ 1 and X> (-1/3) Am I doing something wrong here? Is this the complete solution? Looking at my answer, is there more that I...

Hi everyone, Homework Statement P(X)=Xp-1*(X-1)p*...*(X-n)p j is an integer between 1 and n; x a real beatween 0 and 1. Prove that abs(P(jx))<=(n!)p Homework Equations The Attempt at a Solution I tried to find an inequality for each abs(jx-q) but the problem is that I...

Homework Statement 2 - ((x-3)/(x-2)) ≥ ((x-5)/(x-1)) Homework Equations The Attempt at a Solution I just want to make sure that certain operations that are allowed with fractions in equations are still valid (or not) in inequalities. 2 - ((x-3)/(x-2)) ≥ ((x-5)/(x-1)) Can I...

Homework Statement l lxl - lyl l =< lx-yl Homework Equations n/a The Attempt at a Solution how do i proof this? give me a start please, should i use definition absolute values and consider all of the cases? or use triangle inequality(but i can't figure out how)

Say I have two lists, List1 and List2 containing elements such as words. Some words are common two both List1 and List2. I want to create a distance metric that tells me how far apart the two lists are based on a similarity "score". The similarity score and distance metric are as follows...

does the following inequality holds for every POSITIVE 'x' ? e^{-x}-1\le Cx^{1/4+e} here 'C' and e are positive constants i think that for very very small 'e' the constant must be very BIG but no other hint i find

Homework Statement If f(x) = (x-1)^2 and g(x) = x+1, then g is greater than or equal to f on the set S = {real numbers x : x is between 0 and 3}. Homework Equations g is greater than or equal to f on the set S of real numbers iff for all s in S, g(s) is greater than f(s). The Attempt at a...

Homework Statement Solve the inequality and sketch the graph of the solution on the real number line. Homework Equations |x - a|< or equal to b, b > 0 Let us imagine that the ">" and "<" signs also include "equal to" except for the condition, b > 0, in order to solve this...

Homework Statement Show \sqrt{1+\xi^2}-\xi<1 for \xi>0 Homework Equations The Attempt at a Solution Is this correct way? \sqrt{1+\xi^2}-\xi<1 suppose \sqrt{1+\xi^2}-\xi\geq 1 \sqrt{1+\xi^2}\geq 1+\xi 1+\xi^2 \geq 1+2\xi+\xi^2 0 \geq \xi...

Homework Statement Prove that if b is a positive number, such that a < b, then a/b<(a+1)/(b+1) 2. The attempt at a solution I have tried a few things, attempting to prove it using the real line, and a bunch of other methods but have had no success. I Would greatly appreciate it if u...

The violation of Bell's inequality is often said to imply that either there exists non-locality or there are no hidden variables. In actual experiments it is consistenly found that the inequality is violated by precisely the amount predicted by quantum theory. But quantum theory denies both...

Homework Statement Let f(x) = (1/2+x) log [(1+x)/x] + (3/2-x) log [(1-x)/(2-x)] where log is natural logarithm and 0 < x <= 1/2. Show that f(x) >= 0 for all x.Homework Equations The only inequality that I can think of is the log sum inequality: http://en.wikipedia.org/wiki/Log_sum_inequalityThe...

Homework Statement Solve the inequality, \frac{1.8^n}{n!}<0.201 Homework Equations The Attempt at a Solution some hints?

Homework Statement Explain why the Direct Comparison Test allows us to use the inequality Ln n < n^(1/10) even though it is not true for a great many n values. Homework Equations The Attempt at a Solution I looked at the graphs of Ln (n) vs. n^(k)

To proove the inequality: \left | \int_a^b f(t) dt \right | \le \int_a^b | f(t) | dt for complex valued f, use the following: \textrm{Re}\left [ e^{-\iota\theta}\int_a^b f(t) dt\right ] = \int_a^b \textrm{Re}[e^{-\iota\theta}f(t)] dt \le \int_a^b | f(t) | dt and then if we set: \theta =...

Homework Statement Prove the Inequality for the indicated integer values of n. n!>2^n,n\geq4 Homework Equations The Attempt at a Solution For n=4 the formula is true because 4!>2^4 Assume the n=k k!>2^k Now I need to prove the equation for k+1 I can multiply both sides by 2 and have...

Homework Statement Find all numbers x for which: 2x<8Homework Equations The Attempt at a Solution I really haven't been able to figure this one out.

Homework Statement Show that \left( 1 - \frac{\ln n}{kn} \right)^n > \frac{1}{n^{1/k} + 1} holds for all integers n\geq 1 and k\geq 2. The Attempt at a Solution I first tried to find a proof for k=2 by showing that the quotient LHS/RHS goes to 1 and has negative slope everywhere, but...

Need help proving Cauchy Schwarz inequality ... the first method I know is pretty easy \displaystyle\sum_{i=1}^n (a_ix-b_i)^2 \geq 0 expanding this and using the discriminatant quickly establishes the inequality..The 2nd method I know is I think a easier one , but I don't have a clue about...

Homework Statement If |x-1| < 1 then Prove |x^2 -4x + 3| < 3. Homework Equations The Attempt at a Solution proof: Assume |x-1| < 1. Then X has to be between 0 and 2.Because X has to be between 0 and 2 then |x-3|<3,and |x-1||x-3|<3 by multiplication of inequalities...

Homework Statement If f(x) <= g(x) then lim[x->a] f(x) <= lim[x->a] g(x) provided that both of these limits exist. 2. The attempt at a solution I've been able to prove it by contradiction. So I assumed that l = lim[x->a] f(x) > lim[x->a] g(x) = m. Therefore, l - m > 0 and I could...

Homework Statement Prove the AM-HM inequality using the Cauchy-Schwarz Inequality. Homework Equations Cauchy Schwarz Inequality: \[ \biggl(\sum_{i=1}^{n}a_{i}b_{i}\biggr)^{2}\le\biggl(\sum_{i=1}^{n}a_{i}^{2}\biggr)\biggl(\sum_{i=1}^{n}b_{i}^{2}\biggr)\ AM-HM inequality...

Homework Statement Prove that: y^5+y^2-7y+5\geq 0 ,for all y\geq 1 Homework Equations The Attempt at a Solution y^5\geq 1 and y^2\geq 1 => y^5+y^2\geq 2. Also -7y+5\leq -2 , and then?

Homework Statement I saw this in my real analysis textbook and I have been trying to understand how this equation \left | x-c \right |< 1 you can get this: \left | x \right |\leq \left | c \right | + 1 Homework Equations I wanted to know what steps made this possible ...

Homework Statement From Spivak's Calculus Chapter 1: "Suppose that y_1 and y_2 are not both 0, and that there is no number λ such that x_1 = λy_1 and x_2 = λy_2." Then 0<(λy_1 - x_1)^2 + (λy_2 - x_2)^2. Using problem 18 (which involved proofs related to inequalities like x^2 + xy + y^2)...

Homework Statement Prove: xyzw\geq x+y+z+w-3,where x\geq 1,y\geq 1,z\geq 1,w\geq 1 Homework Equations The Attempt at a Solution The only thing i could try is to square both sides but then this leads nowhere. Any ideas??

I'm interested in thing that are norms except for the fact that they satisfy the reverse triangle inequality ||x+y|| \geq ||x|| + ||y||. The obvious example is taking p-norms for 0<p<1. Does anyone know of others or if there's any theory developed on this topic?

Can anybody tell me why \sum_{j=1}^p |x_j-y_j| \leq \left( \sum_{j=1}^p 1\right)^{1/2} \left( \sum_{j=1}^p |x_j-y_j|^2 \right)^{1/2} is true? Thank you!

Homework Statement Let (X,\theta) be a metric space. Take K > 0 and define. \theta : X \cross X \rightarrow \real_{0}^{+}, (x,y)\rightarrow \frac{K\phi(x,y)}{1+K\phi(x,y)} Show that (X,\theta) is a metric space. Homework Equations can someone please check my triangle...

Homework Statement prove the following inequality: Homework Equations \frac{a^4+b^4+c^4}{a^2b^2+a^2c^2+b^2c^2}\geq\frac{2u+3\lambda}{3(u+\lambda)} for all the reals a,b,c different from zero and for all the natural u,λ The Attempt at a Solution induction i think in this case...

For what values of x and , y is the following inequality satisfied: x^2y+y^2x >6 I tried to give a proof and i went as far: xy(x+y)>6 then what?

sinx ≤ x ≤ tanx You need the above inequality to prove that lim x-> 0 of sinx/x = 1, but I've only ever seen it derived geometrically. Is there an analytical proof of the above inequality, from which you can continue the sinx/x proof as normal?

Homework Statement consider two conditions x2-3x-10 < 0 and |x-2| < A on a real number x, where A is positive real number (i) find the range of values of A such that |x-2| < A is a necessary condition for x2-3x-10 < 0 (ii) find the range of values of A such that |x-2| < A is a sufficient...

Homework Statement Sketch the graph of y = 2|x - 1| - 3|x + 1| + 3x + 1, and hence solve the inequality 2|x - 1| - 3|x + 1| + 3x + 1 < 0 Homework Equations None The Attempt at a Solution (Refer to attachment). I don't know where (or if) I made a mistake, cause when I try...