# Thread: Induction Proof of Sum of n positive numbers

1. ## Induction Proof of Sum of n positive numbers

I am not sure if this is the right place, but it does appear in my calculus book.

"
Proove by induction that for all n real, positive numbers a1, a2, a3, ..., an that follow the rule a1a2a3...an = 1, the following expession is true:

$\displaystyle \sum_{i=1}^{n} a_{i} \geq n$

"

I have tried working it out but I couldn't proove the induction step.

I would very appreciate any help with this.

2. ## Re: Induction Proof of Sum of n positive numbers

assuming

$$x y=1 \Longrightarrow x+y\geq 2$$

prove that

$$a b c =1 \Longrightarrow a+b+c \geq 3$$

if we suppose $a\geq b\geq c$ then $a\geq 1$ since otherwise $a b c <1$.

Likewise $c\leq 1$ and so we have the inequality $(a-1)(1-c) \geq 0$

Applying the induction hypothesis to $x=a c$ and $y= b$

we get a second inequality $a c+b \geq 2$

Adding these two inequalities we have

$$a+b+c \geq 3$$