Verified answer

To start off, note that

0 ≤ (|x| - |y|)^2

because |x| - |y| is real and the square of any real number is nonnegative. Expanding the right side gives

0 ≤ |x|^2 - 2|x||y| + |y|^2

Adding |x|^2 + 2|x||y| + |y|^2 to each side gives

|x|^2 + 2|x||y| + |y|^2 ≤ 2|x|^2 + 2|y|^2

Factoring both sides gives

(|x| + |y|)^2 ≤ 2(|x|^2 + |y|^2)

Taking the square root of each side (which preserves the inequality because the square root function is increasing) gives

|x| + |y| ≤ (√2) (√(|x|^2 + |y|^2))

But √(|x|^2 + |y|^2) = √(x^2 + y^2) = |z|, so

|x| + |y| ≤ (√2) |z|,

as desired.