Triangle inequality Definition and 101 Threads

Is it true in general that: |\int f(x)dx| < \int |f(x)|dx Not sure if "Triangle Inequality" is the right word for that, but that seems to be what's involved.

Homework Statement d(x,y)=(a|x_1-y_1|^2+b|x_1-y_1||x_2-y_2|+c|x_2-y_2|^2)^{1/2} where a>0, b>0, c>0 and 4ac-b^2<0 Show whether d(x,y) exhibits Triangle inequality? Homework Equations (M4) d(x,y) \leq d(x,z)+d(z,y) (for all x,y and z in X) The Attempt at a Solution I...

I was wonder what conditions a and b have to be for each inequality in order to satifsy the equality?

Pretty much knows the triangle inequality. \left| a + b \right| \le \left| a \right| + \left| b \right| I was reading a source which asserted the following extension of the triangle inequality: \left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right) This is...

[FONT="Arial Black"]hi every body show if Xn→x then lXnl→lxl hint use trangle inequality 2/ show if lXnl→0 then Xn →0 show by example that lXnl fore all n in N MAY CONVERGE and Xn may not converge

Homework Statement The question says to find a proof for Cauchy's Inequality and then the Triangle Inequality. This is an elementary linear algebra class I'm doing, so I can't use inner products or anything. Homework Equations The Attempt at a Solution I got the proofs using algebra, but I'm...

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...

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

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...

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?

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: If the limit inf as k goes to infinity of abs(ak+1 / ak) > 1 then the sum from 1 to infinity of ak diverges Homework Equations The Attempt at a Solution So far I have this: Suppose lim inf abs(ak+1/ak) >1 then, there exists an r such that lim inf...

Homework Statement Prove: If the limit inf as k goes to infinity of abs(ak+1 / ak) > 1 then the sum from 1 to infinity of ak diverges Homework Equations The Attempt at a Solution So far I have this: Suppose lim inf abs(ak+1/ak) >1 then, there exists an r such that lim inf...

Hi Guy's I was wondering If anyone new of a good link about the P-adic metric Strong triangle inequality. I am trying to constuct a proof to show that the p-adic is a metric space. Must the proof use the Strong triangle inequality ie d(a,c)\leq max{d(a,b),d(b,c)} or can it use the...

I was wondering if anyone knew of a text or website that has a very indepth explanation of the triangle inequality and its variations really just how to apply it with limit proofs. I mean i understand it in terms of a triangle as no one side can be longer than the sum of the two other sides...

Hey folks, can someone quickly check my algebra? Given: d(x,A) \leq d(x,y) + d(y,A) To show: \left|d(x,A) - d(y,A) \right| \leq d(x,y) Proof: from given, d(x,A) - d(yA) \leq d(x,y); and -d(x,A) + d(y,A) \geq -d(x,y); \Rightarrow d(y,A) - d(x,A) \geq -d(x,y)...

Homework Statement Use Schwarz inequality on \bar{v} \bullet \bar{w} to prove: ||\bar{v} + \bar{w}||^2 \leq (||\bar{v}|| + ||\bar{w}||)^2 Homework Equations Schwarz inequality: |\bar{v} \bullet \bar{w}| \leq ||\bar{v}|| ||\bar{w}|| The Attempt at a Solution The way I...

Homework Statement Use the triangle inequality to show: \left|z^2+3\right| \leq(12) for \left|z\right|=3 where z is a complex number Homework Equations triangle inequality: \left|z_1+z_2\right|\leq \left|z_1\right|+\left|z_2\right| The Attempt at a Solution I understand the...

Homework Statement Prove that any side of a triangle is less than or equal to the sum of the other two sides of the triangle (using components). Homework Equations root [(x3-x1)^2 + (y3-y1)^2] <= root [(x2-x1)^2 + (y2-y1)^2] + root [(x3-x2)^2 + (y3-y2)^2] The Attempt at a Solution...

http://math.ucsd.edu/~wgarner/math4c/derivations/other/triangleinequal_files/eq0007S.gif Why did they introduce the <= sign? I cannot think of any numbers that would violote the =. So why introduce the <?

I should know how to do this, but I just can't figure it out. Should be a piece of cake. How do you prove, for x,y\in \mathbb{R}^n: \left| (||x|| - ||y||) \right| \leq ||x-y||

...to prove Pythagoras' theorem? Or is the triangle inequality considered a corollary of Pythagoras theorem (so it can't be used to prove it)? I have an assingment question that asks you to prove Pythagoras' theorem, so I just wanted to know if the inequality can be used in this case. Thanks...

In lecture in my real analysis course the other day we were proving that absolute convergence of a series implies convergence. Our professor started off by showing us the wrong way to prove it: \left| \sum_{k=1}^\infty a_k \right| \leq \sum_{k=1}^\infty \left| a_k \right| < \epsilon Then he...

Homework Statement assuming an and bn are cauchy, use a triangle inequality argument to show that cn= | an-bn| is cauchy Homework Equations an is cauchy iff for all e>0, there is some natural N, m,n>=N-->|an-am|<e The Attempt at a Solution I am currently trying to work backwards on this one...

Homework Statement If X is a metric space and A is a subset of X, show that |d(x,A) - d(y,A)|\leq d(x,y) for any x,y in X.Homework Equations Triangle inequality: |d(x,z) - d(y,z)|\leq d(x,y), d(x,y)\leq d(x,z)+d(z,y)The Attempt at a Solution Fidling around with the triangle inequality...

in order for me to understand this theorim do i have to think of the vectors as complex?? what the values of x1 ,y1 ,x2 ,y2 \langle x , y \rangle = x_1^* \cdot y_1 + x_2^*\cdot y_2 + \ldots did i get the structure correctly x=(x1,y1) y=(y1,y2) each one of...

I have been trying the proof of the reverse triangle inequality for n+1 real numbers: |x-y1-y2-y3-...-yn| \geq | |x| - |y1| - |y2| - |y3| - ...-|yn| | I know the proof of the reverse triangle inequality for 2 real numbers and the triangle inequality for n numbers. can somebody help ?

Hi guys! There is one question after which I have been for hours. I need to prove that: tan C > b/c in an acute angled triangle. I cannot get started. I tried to convert b/c to sin B/sin C but no useful results. .. and so on. Please help me as soon as possible!

My professor said this was the triangle inequality. We're to use mathematical induction to prove it. I've gotten some work done, and after "proving" it, it just seems to easy. :| http://answerboard.cramster.com/advanced-math-topic-5-321495-0.aspx" .

Homework Statement Using the fact that |z(1)-z(2)| is the distance between two points z(1) and z(2) , give a geometric argument that a)|a-4*i| + |z+4*i| =10 represents an ellipse whose foci are (0,4) and(0,-4). Homework Equations Triangle inequality equation; distance formula...

Homework Statement Show for nonnegative x,y,z that (x+y+z) \sqrt{2} \leq \sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{x^2+z^2} My book says the answer comes from apply the triangle inequality to (x+y+z,x+y+z)=(x,y)+(y,z)+(x,z). I don't see what they mean by that at all. HOW do you apply the triangle...

Is it possible to write the triangle inequality in 2 ways: 1. d(x,z) <= d(x,y) + d(y,z) 2. d(x,y) <= d(x,z) + d(z,y) ? If so, are these the only 2 ways? Thanks

Homework Statement Prove that |x+y|</|x|+|y| Homework Equations The Attempt at a Solution Assume that x and y are real numbers. |5+2|</|5|+|7| 7</7 |-5-2|</|-5|+|-2| 7</7 I know that it is true by testing different numbers, but I'm not sure how to prove it. Could someone...

Homework Statement I don't know if I'm posting in the right area, but here is the question: From the inequality: la dot bl= lal lbl lcos(theta)l is less than or equal to lal lbl Deduce the triangle inequality: la + bl is less than or equal to lal +lbl Homework Equations...

To make it clear, I understand the theorem and several proofs of this theorem but the most basic one is not making sense. Thm: |a+b|<or=|a|+|b| Proof: (a+b)^2 = a^2+2ab+b^2 < or = |a|^2 + 2|a||b|+|b|^2 = (|a| + |b|)^2 Taking the square root of both sides and remember that |x|=square...

Homework Statement show |(|z|-|z'|)|<=|z-z'| The Attempt at a Solution I used z=a+ib and z'=a'+ib' and ended up with the reverse inequality to the above by proving (ab'-ba')^2>=0 hence the reverse of the sign above. Where have I gone wrong?

Homework Statement Show that if |a-5| < 1/2 and |b-8| < 1/2 then |(a+b)-13| < 1. Hint: use the triangle inequality. Homework Equations Triangle Inequality: |a+b| <= |a|+|b|The Attempt at a Solution I really don't know how to use the triangle inequality so I was hoping someone could clear...

I am a beginner at trying to prove or disprove inequalities. In an attempt to improve on this skill I found some problems that I would like to work on. Now, I know many of you may be able to look at this and think of a solution, but please refrain from posting it, but some advice and methods...

What's so special about the Triangle Inequality? abs(x + y) <= abs(x) + abs(y) I have learned it in two or three units, but it seems too obvious to be given a special name. What have I missed? EDIT: I screwed up the tex here... if someone can fix it that would be cool

can someone explain this proof please, I added a star to the inequalities I don't see/understand. if | | is a norm on a field K and if there is a C > 0 so that for all integers n |n.1| is smaller than or equal to C, the norm is non archimedean (ie the strong triangle inequality is true)...

"Use the triangle inequality to prove that: ||\vec{a}|| - ||\vec{b}|| \le ||\vec{a} - \vec{b}||" I can start from that expression and prove it true using the Cauchy-Shwarz inequality but I don't think that's what's asked. Any hints?

Hello all, I am having some difficulties with a question. Hope you guys can help shine some light on the situation. Use the triangle Inequality and the fact that 0< |a| < |b| + => 1/|b| < 1/|a| to establish the following chains of inequalities. |x-2 / x^2+9| "Is less than and equal to"...

Edit: Nevermind i got it, thanks anyway for complex numbers z1 and z2 prove that |z_{2}| - |z_{1}| \leq |z_{2} - z_{1}| the left hand side becomes \sqrt{x_{2}^2 + y_{2}^2} - \sqrt{x_{1}^2 + y_{1}^2} the right hand side becomes \sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2} now i tried...

what does a triangle have to do with triangle inequality, and what does a paralllelogram have to do with parallelogram equality? @@a

I know that the triangle inequality is lx+y|<= |x| +|y| Does this hold for a summation of say Sigma (|xn+yn|) <= sigma|xn| + sigma|yn| for n=0 to infinity? would this also work for sup|x+y| ??

I have proven the triangle inequality starting with ||a+b||^2 and using the Schwartz Inequality. However, the next part of the problem says: "Show that the Triangle Inequality is an equality if and only if |a>=alpha|b> where alpha is a real positive scalar." It must be proved in both...

I must prove \lim_{x\rightarrow 3\\} x^2 = 9 I get this... \mid x+3\mid\mid x-3\mid < \epsilon if 0< \mid x-3 \mid < \delta then it says with the triangle inequality we see that \mid x+3\mid = \mid (x-3)+6\mid \le \mid x-3\mid +6 therefore if 0< \mid x-3 \mid < \delta , then...

Hi everyone I need some help proving Pedoe's Inequality for two triangles, which states that a_{1}^2(b_{2}^2+c_{2}^2-a_{2}^2) + b_{1}^2(c_{2}^2 + a_{2}^2 - b_{2}^2) + c_{1}^2(a_{2}^2 + b_{2}^2 - c_{2}^2) \geq 16F_{1}F_{2} where (a_{1},b_{1},c_{1}) and (a_{2},b_{2},c_{2}) are the sides...

Does the triangle inequality hold true for three vectors that says the norm(u+v+w)<=norm(u)+norm(v)+norm(w)...true or false