Askhwhelp said:Prove |x|+|y| ≤ |x+y| + |x-y| for all real numbers x and y.
Some ideas I have is let a = x+y and b = x-y and apply triangle inequity
Could anyone give me some direction?
Thanks
Askhwhelp said:let a = x+y and b = x-y.
|2x| = |a+b| <= |a| + |b|
|2y| = |a- b| <= |a| + |b|
So |2x| + |2y| <= 2(|a| + |b|)
Divide both sides by 2, we get
|x|+ |y| <= |a| + |b|
Is this right way?
Thanks
The triangle inequality in real analysis is a fundamental property that states that the sum of any two sides of a triangle must be greater than the length of the third side. In mathematical notation, it can be written as |a + b| ≥ |a| + |b|, where a and b are the sides of the triangle. This inequality is a crucial concept in real analysis and has many applications in geometry and other branches of mathematics.
The triangle inequality is used in many ways in real analysis, including proving the convergence or divergence of sequences and series, establishing the existence and uniqueness of solutions to differential equations, and defining and analyzing metric spaces and normed vector spaces. It is also used to prove inequalities for various mathematical objects, such as inequalities for integrals and derivatives.
The triangle inequality is significant because it is a powerful tool for proving many important results in real analysis. It is a fundamental property that is used extensively in various branches of mathematics, including calculus, geometry, and functional analysis. Additionally, it provides a way to quantify the distance between two points in a metric space and is essential for defining and studying various types of inequalities.
Yes, the triangle inequality can be extended to higher dimensions. In fact, it is a generalization of the triangle inequality in real analysis. In higher dimensions, the sum of the lengths of any two sides of a polygon must be greater than the length of the third side. This concept is crucial in the study of geometric objects in higher dimensions, such as polytopes and convex sets.
No, there are no exceptions to the triangle inequality in real analysis. It is a universal property that holds for all triangles in any metric space. However, in some cases, the triangle inequality can be weakened to an equality. For example, in a degenerate triangle where all three sides are collinear, the triangle inequality becomes an equality. But in general, the triangle inequality holds true for all triangles.