<s>[INST] $\frac{a}{\sqrt{a+b}} + \frac{b}{\sqrt{b+c}} + \frac{c}{\sqrt{c+a}} > \sqrt{a+b+c}$ is true for positive a,b,c How do you prove that for all positive $a,b,c$ this formula holds true:
\begin{equation*}
\frac{a}{\sqrt{a+b}} + \frac{b}{\sqrt{b+c}} + \frac{c}{\sqrt{c+a}} > \sqrt{a+b+c}?
\end{equation*}
Any help will be invaluable.
 [/INST] Since
$$a+b+c=\sqrt{(a+b+c)^2}=\sqrt{a+b+c}\ \sqrt{a+b+c}\iff \dfrac{a+b+c}{\sqrt{a+b+c}}=\sqrt{a+b+c}$$
and
$$\dfrac{a}{\sqrt{a+b}}>\dfrac{a}{\sqrt{a+b+c}}$$
$$\dfrac{b}{\sqrt{b+c}}>\dfrac{b}{\sqrt{a+b+c}}$$
$$\dfrac{c}{\sqrt{c+a}}>\dfrac{c}{\sqrt{a+b+c}}$$
we have
$$\dfrac{a}{\sqrt{a+b}}+\dfrac{b}{\sqrt{b+c}}+\dfrac{c}{\sqrt{c+a}}>\dfrac{a+b+c}{%
\sqrt{a+b+c}}=\sqrt{a+b+c}.$$
</s>