How to prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$?

Question

How do I prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? for $x, y$ positive?

This should be easy, but I'm not seeing how. A hint would be appreciated.

Answer 1

For positive $x, y$, we have:

$$\sqrt{x + y} \leq \sqrt{x} + \sqrt{y} \iff \left(\sqrt{x + y}\right)^2 \leq \left(\sqrt x + \sqrt y\right)^2 \iff \color{blue}{\bf x + y \leq x + y + 2\sqrt{xy}}$$

What can you conclude about the leftmost "inequality", given its equivalence to the $\color{blue}{\bf rightmost\;inequality}$?

Answer 2

Hint: For positive numbers $a$ and $b$, $$a\leq b\iff a^2\leq b^2.$$

Answer 3

Putting everything together, suppose $x,y>0$. Then $0\le2\sqrt{xy}$. Hence: $$ \sqrt{x+y} = \sqrt{x+0+y} \le \sqrt{x+2\sqrt{xy}+y} = \sqrt{(\sqrt{x}+\sqrt{y})^2} = \sqrt{x}+\sqrt{y} $$ as desired. Note that this relied on the fact that $f(x)=\sqrt{x}$ is monotonically increasing.

Answer 4

Draw a right triangle with the two sides making the $90^\circ$ angle of length $\sqrt{x}$ and $\sqrt{y}$. Then, by the Pytagorean Theorem, the Hypotenuse is $\sqrt{x+y}$.

Since the sum of two edges exceeds the third edge you get

$$\sqrt{x+y} < \sqrt{x}+\sqrt{y} \,.$$