given positive real numbers x,y how to demonstrate that is defined as the maximum of x and y?

Question

Is it true that given positive real numbers $x,y$, then we have that

$$ \sqrt{x^2 + y^2} \geq \max\{ x, y \} $$

I cant find a counter-example although it seems it is true... Any comments?

Answer 1

We know that

$${x^2} + \underbrace {{y^2}}_{ \ge 0} \ge {x^2}$$

$$\underbrace {{x^2}}_{ \ge 0} + {y^2} \ge {y^2}$$

So, taking the square root on both sides of each expression, we get

$$\sqrt {{x^2} + {y^2}} \ge \left| x \right| \ge x$$

$$\sqrt {{x^2} + {y^2}} \ge \left| y \right| \ge y$$

Thus

$$\sqrt {{x^2} + {y^2}} \ge \max \left\{ {x,y} \right\}.$$

Answer 2

Yes. To see this, note that $$\tag{1}\sqrt{x^2+y^2}\geq \sqrt{x^2}=x$$ since $x$ is positive. Similarly, we have $$\tag{2}\sqrt{x^2+y^2}\geq \sqrt{y^2}=y$$ since $y$ is positive. Combining $(1)$ and $(2)$, we have $$\sqrt{x^2+y^2}\geq\max\{x,y\}.$$