# Thread: proving for real positive numbers

1. ## proving for real positive numbers

how can i do?
For a,b,c real positive numbers, show that ;

(a+b+c)(1/a + 1/b + 1/c)>= 9

is there any theorem to use?
i dont know theorems based on this..
can u help me with this?

2. Using the AM-GM-HM inequality,

$\displaystyle AM = \frac{a+b+c}{3}$

$\displaystyle HM = \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$

$\displaystyle AM \geq GM \geq HM$

$\displaystyle AM \geq HM$

$\displaystyle \frac{a+b+c}{3} \geq \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$

$\displaystyle (a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}) \geq 9$

3. ## thank u very much..

i don't even know the fact that .
thankx for ur help alexmahone.. i really appreciate it..

$\displaystyle (a - b )^2 \geq 0, \mbox{ so } a^2 + b^2 - 2ab \geq 0$.
$\displaystyle \ a^2 + b^2 \geq 2ab \mbox{ so } \frac{a^2 + b^2}{ab} \geq 2 \mbox{ ( given that } a \mbox{ and } b$ are positive numbers).
So, multiply out the LHS of your inequality and collect up terms. The two terms involving $\displaystyle a \mbox{ and } b$ can be combined to produce the fraction $\displaystyle ( a^2 + b^2 ) / ab$, so you can make use of the result obtained earlier.
Do the same for the terms involving $\displaystyle b \mbox{ and } c$ and those involving $\displaystyle c \mbox{ and } a$ , and the result follows.