Hi, I have the following problem which I think I have solved correctly (it would be great if you could check it), but I don't know how to answer to "simple" questions about my proof... any help would be very much appreciated, thanks!

The problem says:

"Let t>-1 be given with t≠0. Show by induction that for n≥2 we have

."

And here are the two questions that I don't know:

"Where is the condition t>-1 used in your proof? Is the result true if instead we had -2<t<-1?"

My solution to the first bit is:

Let .

Proof by induction on n.

Base n=0. Then

Induction. Suppose that there is a number k (k≥2) such thaqt for any t>-1 (t≠0). We need to show that (1+t)^{k+1}>1+(k+1)t for any t>-1, t≠0.

. By the induction hypothesis, , so we have:

.

Since and .

So

And so the result holds for n=0 and hence for all n≥2 and t>-1,t≠0.