Problem: Show that for any $a, b \in ℝ ^n: \|a + b\|\geq\|a\|-\|b\|$
I have a feeling that we can use the triangle inequality here somehow. I am not sure how to start this proof?
2026-02-23 15:06:14.1771859174
How to prove the following using triangle inequality?
50 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in PROOF-WRITING
- how is my proof on equinumerous sets
- Do these special substring sets form a matroid?
- How do I prove this question involving primes?
- Total number of nodes in a full k-ary tree. Explanation
- Prove all limit points of $[a,b]$ are in $[a,b]$
- $\inf A = -\sup (-A)$
- Prove that $\sup(cA)=c\sup(A)$.
- Supremum of Sumset (Proof Writing)
- Fibonacci Numbers Proof by Induction (Looking for Feedback)
- Is my method correct for to prove $a^{\log_b c} = c^{\log_b a}$?
Related Questions in TRIANGLE-INEQUALITY
- Norm squared inequality
- Gre question : finding length of sides of a triangle given the longest side length and each side has an integer length
- Successive prime numbers and triangles
- Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
- For $a$, $b$, $c$ the sides of a triangle, show $ 7(a+b+c)^3-9(a+b+c)\left(a^2+b^2+c^2\right)-108abc\ge0$
- If $\vert x - a \vert < \frac{1}{2}\vert a\vert$, then $\frac{1}{2}\vert a\vert < \vert x \vert$?
- When does equality hold in the triangle inequality?
- Prove length relationship of median and sides in triangle using triangle inequality
- Prove $a^2 + b^2 \geq 2ab$ using Triangle Inequality
- Guessing the third side of the triangle from the given two sides
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Note that $(a+b)+(-b)=a$. Therefore by the triangle inequality as you suggest, $$\|a\|=\|(a+b)+(-b)\|\leq\|a+b\|+\|-b\|=\|a+b\|+\|b\|.$$ Rearranging gives your desired result.