How do you prove triangular inequalities?

Triangle Inequality Proof

1. Since the sum of any two sides is greater than the third, then the difference of any two sides will be less than the third.
2. The sum of any two sides must be greater than the third side.
3. The side opposite to a larger angle is the longest side in the triangle.

What is the formula of triangle inequality?

The triangle inequality states that: For any triangle the length of any two sides of the triangle must be equal to or greater than the third side. Sometimes seen as: |X+Y|≤|X|+|Y| The inequality works not only if X and Y are both real numbers (scalars), but also if X and Y are vectors (of the same dimension).

How do you prove inequality holders?

Proof of Hölder’s inequality

1. If ||f ||p = 0, then f is zero μ-almost everywhere, and the product fg is zero μ-almost everywhere, hence the left-hand side of Hölder’s inequality is zero.
2. If ||f ||p = ∞ or ||g||q = ∞, then the right-hand side of Hölder’s inequality is infinite.

What is triangle inequality statement?

triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c. In essence, the theorem states that the shortest distance between two points is a straight line.

What is triangle inequality theorem 1?

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

What is power mean inequality?

The Power Mean Inequality states that for all real numbers and , if . In particular, for nonzero and. , and equal weights (i.e. ), if , then. Considering the limiting behavior, we also have , and .

How to prove CW and triangle inequality?

1 Start both proofs with the fact that a vector dotted with itself is greater than or equal to 0 2 for CW substitute vector = x -t y, for triangle inequality vector = x + y 3 for CW, after dotting x -t y with itself let t = ( x. y )/ ( y. y ), for triangle ineq. 4 after rearranging both sqrt both sides

What is triangle inequality in math?

Triangle Inequality. Theorem: In a triangle, the length of any side is less than the sum of the other two sides. So in a triangle ABC, |AC| < |AB| + |BC|. (Also, |AB| < |AC| + |CB|; |BC| < |BA| + |AC|.) This is an important theorem, for it says in effect that the shortest path between two points is the straight line segment path.

How do you prove that the equality is true?

Proof: First we prove that the equality is true if B is between A and C.  Choose a ruler on the line AB; then the 3 points correspond to numbers a, b, c and either a <  b < c or c < b < a. Suppose a  <  b < c.

Is the triangle inequality an axiom in the metric space?

Additionally, the triangle inequality is an axiom in metric spaces, but it is not axiomatic that $M = (\\mathbb{R},|\\cdot|)$ is a metric space, hence we need to prove the triangle inequality in this case by first principles to demonstrate that $M$ is truly a metric space.