Which Inequality? $\sum_{i=1}^{n}\frac{1}{x_i}\geqslant1$

Question

Can we prove this? What is the name for this inequality?

Or is there a counter example?

$$\forall\;x_i\gt0\mid\prod_{i=1}^{n}x_i=1,\;\;\;\;\;\sum_{i=1}^{n}\dfrac{1}{x_i}\geqslant1.$$

Random facts with short proofs usually won't have their own name. — Zarrax, Jun 13 '14 at 16:07

score 6 · Accepted Answer · answered Jun 13 '14 at 16:04

6

If $\prod x_i=1$ then some $x_{i_0}\leq 1$ and thus $\sum\frac{1}{x_i} \geq\frac{1}{x_{i_0}}\geq 1$.

answered Jun 13 '14 at 16:04

Thomas Andrews

177,126

score 5 · Answer 2 · answered Jun 13 '14 at 16:08

5

Thomas Andrews has answered the question as stated, but, in fact, you have $\sum_{i=1}^{n} \frac{1}{x_{i}} \geq n$ under those circumstances, by the arithmetic-geometric mean inequality.

answered Jun 13 '14 at 16:08

Geoff Robinson

24,035

Indeed, just look at the arithmetic and geometric means of $\dfrac1{x_i}$ (+1) – robjohn Jun 13 '14 at 17:01

Which Inequality? $\sum_{i=1}^{n}\frac{1}{x_i}\geqslant1$

2 Answers2