Are there numbers x, y such that |x + y| < |x| + |y|?

To prove : |x + y| < |x| + |y|

We know that, |x| and |y|

Therefore, 2|x||y|

Adding x² + y² to both sides,

We have, x² + y² + 2|x||y| x² + y² + 2xy

⇒ |x|² + |y|² + 2|x||y| x² + y² + 2xy

⇒ (|x| + |y|)² (x + y)²

⇒ (|x| + |y|)² (|x + y|)²

⇒ |x| + |y| |x + y|

We can also say that |x| + |y| > |x + y|

Therefore, this inequality holds true for all x and y.