1. Suppose that x 1 , x 2 , …, x n are real numbers. Prove that |x 1 + x 2 + … + x n | ≤ |x 1 | + |x 2 | + … + |x n | . 2. Let P = {x ∈ R: x > 0}. Show that P satisfies the following: (a) If x, y ∈ P,...

1. Suppose that x₁, x₂, …, x_n
are real numbers. Prove that

|x₁
+ x₂
+ … + x_n| ≤ |x₁| + |x₂| + … + |x_n| .

2. Let P = {x ∈ R: x > 0}. Show that P satisfies the following:

(a) If x, y ∈ P, then x + y ∈ P.

(b) If x, y ∈ P, then x ⋅ y ∈ P.

(c) For each x ∈
, exactly one of the following three statements is true: x ∈ P, x = 0, − x ∈ P.