Use induction on k to prove that if 𝐱1, …, 𝐱k ∈ℝ^n, then 𝐱1+⋯+𝐱k≤𝐱1+⋯+𝐱k.

Name: Problem 13, Chapter 1: Vector Calculus
Uploaded: 2022-01-26T04:56:00-08:00
Duration: 2 min 27 s
Channel: Nick Derr
Description: Use induction on k to prove that if 𝐱1, …, 𝐱k ∈ℝ^n, then 𝐱1+⋯+𝐱k≤𝐱1+⋯+𝐱k.

step 1 In this problem, we want to use induction to show that the sum of k vectors x1 plus x2 plus all

Use induction on k to prove that if 𝐱1, …, 𝐱k ∈ℝ^n, then...

Use induction on $k$ to prove that if $\mathbf{x}_{1}, \ldots, \mathbf{x}_{k} \in \mathbb{R}^{n},$ then
$$\left\|\mathbf{x}_{1}+\cdots+\mathbf{x}_{k}\right\|\leq\left\|\mathbf{x}_{1}\right\|+\cdots+\left\|\mathbf{x}_{k}\right\|$$.

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Key Concepts

Triangle Inequality

The triangle inequality is a fundamental property in normed vector spaces stating that the norm of the sum of two vectors is at most the sum of their norms. This principle, originally inspired by geometric interpretations, serves as the cornerstone for bounding the distance in various mathematical contexts and plays a crucial role in the structure of metric spaces.

Properties of Norms

Norms are functions that assign a non-negative length or size to vectors in a vector space and satisfy specific properties: non-negativity, scalar multiplication, the triangle inequality, and that the norm is zero if and only if the vector is the zero vector. These properties ensure that norms behave in a consistent and predictable manner which is foundational to many areas in analysis and geometry.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of propositions. It involves two main steps: verifying the base case is true and then proving that if the statement holds for an arbitrary case (typically n = k), it must also hold for the next case (n = k+1). This method is particularly useful for assertions formulated over the natural numbers.

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