Inequality problem: Application of Cauchy-Schwarz inequality Let $a,b,c \in (1, \infty)$ such that $ \frac{1}{a} + \frac{1}{b} + \frac{1}{c}=2$. Prove that:
$$
\sqrt {a-1} + \sqrt {b-1} + \sqrt {c-1} \leq \sqrt {a+b+c}.
$$
This is supposed to be solved using the Cauchy inequality; that is, the scalar product inequality.

We apply the Cauchy Schwarz Inequality to the vectors $x = \left({\sqrt {\dfrac{a-1}{a}} , \sqrt {\dfrac{b-1}{b}} , \sqrt {\dfrac{c-1}{c}}}\right) $ and $y = \left({\dfrac{1}{\sqrt{bc}},\dfrac{1}{\sqrt{ac}}, \dfrac{1}{\sqrt{ab}} }\right)$ in $\Bbb R^3$. 
Then, $x \cdot y \le \lVert x\rVert \lVert y\rVert$ yields, 
$$ \dfrac{\sqrt{a-1} + \sqrt{b-1} + \sqrt{c-1}}{\sqrt{abc}} \le \sqrt{\dfrac{a-1}{a} + \dfrac{b-1}{b} + \dfrac{c-1}{c}} \times \sqrt{\dfrac{1}{bc} + \dfrac{1}{ac} + \dfrac{1}{ab}}$$
Now manipulating the $\sqrt{abc}$ across the sign we get 
$$ \sqrt{a-1} + \sqrt{b-1} + \sqrt{c-1} \le \sqrt{\dfrac{a-1}{a} + \dfrac{b-1}{b} + \dfrac{c-1}{c}} \times \sqrt{a+ b+c} $$
So, $$ \dfrac{\sqrt{a-1} + \sqrt{b-1} + \sqrt{c-1}}{\sqrt{a+ b+c}} \le \sqrt{\left({1 - \dfrac 1 a}\right) + \left({1 - \dfrac 1 b}\right) + \left({1 - \dfrac 1 c}\right) }$$
$$ \dfrac{\sqrt{a-1} + \sqrt{b-1} + \sqrt{c-1}}{\sqrt{a+ b+c}}  \le \sqrt{3 - \left({\dfrac 1 a + \dfrac 1 b + \dfrac 1 c}\right)} = \sqrt{3-2} = 1$$
$\mathscr{Q.E.D}$
