Prove that $\sqrt{a}\leq\frac{1+a}{2}$

I have a math homework where it's being asked to prove that :

$$\forall a \geq 0,\sqrt{a}\leq\frac{1+a}{2}$$

However, I don't have any idea how I should start this one...

Any idea ?

Solution 1:

Try expanding $$ (\sqrt a - 1)^2 \geq 0 $$

More generally, let $a,b\geq0$.

$$(a-b)^2 \geq0$$

$$a^2-2ab+b^2 \geq0$$

$$a^2+2ab+b^2 \geq 4ab$$

$$(a+b)^2 \geq 4ab$$

$$\left(\frac{a+b}{2}\right)^2 \geq ab$$

$$\frac{a+b}{2} \geq \sqrt{ab}$$