I'm stuck on this proof. I'm not sure if I've done something wrong, or if I'm just not seeing the next step. Any help would be appreciated.

Let a1, a2, ..., an be positive real numbers such that a1*a2*...*an=1. Prove by induction that (1+a1)(1+a2)...(1+an) >= 2^n. The hint for the problem says to try a reduction by introducing another variable that replaces two specially chosen numbers from the sequence.

The base case is simple enough. On the induction step, I assume a1*a2*...*an=1 implies (1+a1)(1+a2)...(1+an) >= 2^n. Then I let z=an*an+1. Thus, a1*a2*...*an-1*z implies (1+a1)(1+a2)...(1+an-1)(1+z) >= 2^n. I then multiplied both sides of the inequality by 2, to get:

(1+a1)(1+a2)...(1+an-1)(1+z)(2) >= 2^(n+1)

At this point, I figured I could try to show that (1+an)(1+an+1)>=(1+z)*2, which would lead to (1+a1)(1+a2)...(1+an)(1+an+1 >= 2^(n+1), but that didn't quite seem to work out.

Any thoughts?

Thanks!