Combining the Binomial Theorem with the sequence

, we get the following rational inequality…

**Claim.** For it holds that

**Proof.**

Applying the binomial theorem,

Look at the factors of the product above: fix and define , clearly for any , so is a monotonically increasing function of . This imply provided that .

Also , thus

Now for every it holds , this imply

.

**Q.E.D.**