- #1

- 5

- 0

## Homework Statement

Prove that |a + b| ≤ |a| + |b|.

## Homework Equations

|a| = √a

^{2}

## The Attempt at a Solution

Since |a| = √a

^{2}, then

|a + b| = √(a + b)

^{2}= √(a

^{2}+ 2ab + b

^{2}) = √a

^{2}+ √b

^{2}+ √(2ab) = |a| + |b| + √(2ab).

And since the square root of a negative number is not defined, then 2ab must be ≥ 0.

This proves the theorem that |a + b| ≤ |a| + |b|.